I've been very happy with the two sets and boards I got from Regency Chess. Good service too.
Help with reputable suppliers

Chess Sets for Club & Tournament Play...
https://www.chess.com/blog/RussBell/chess-sets-for-club-tournament-play
https://www.chess.com/blog/RussBell
Note that "ebonized" wood pieces are not "painted". They are typically boxwood which has been STAINED black. The overwhelming majority of black wood pieces sold by reputable dealers are ebonized. In spite of what some folks may say to the contrary, there is typically nothing wrong (i.e., inferior) with ebonized pieces. The dealers cited in my article as well as those mentioned above by @DesperateKingWalk are reputable.

@DesperateKingWalk -
I stand by what I said. The overwhelming majority of "ebonized" pieces sold by reputable dealers are stained black, not painted. That is what the term "ebonized" implies. If they are painted they are, technically, not "ebonized".
ChessHouse is a reputable chess dealer in the state of Washington, USA. Here they explain "ebonized"...
https://www.chesshouse.com/collections/wood-chess-pieces-ebonized
Feel free to call them and ask to speak with the owner, Raphel Neff. He is a good guy, very knowledgeable and will be happy to help you.

@DesperateKingWalk -
I qualified my statement by saying the overwhelming majority. That doesn't rule out exceptions.

Or you can communicate with the reputable source retailer/supplier (not a third party like Amazon), and resolve those questions regarding painted vs stained ebonized pieces with them. However understand that "painted" pieces are, strictly speaking, not "ebonized" which refers to the process of staining the wood black.
Thank you to everyone on your input and advice so far. It has given me a wealth of options to explore to find my set!
On the case of ebonized wood, I'm likely to steer clear for this set for two reasons:
- I don't want to risk it being painted even if the majority are genuinely ebonized
- I do love being able to see the wood's natural beauty:
- I've done a little lathing and one of my favourite parts was finding an interesting pattern that became a feature
- I love the set at my club that has a singular pawn with an interesting pattern - it would make the natural look worth it

Amazon is a third party seller, not the source supplier. But it is true that Amazon typically makes it hassle free to return products that are clearly defective. However if, prior to purchase, you want to know whether the black chess pieces are painted versus ebonized, Amazon representatives will typically not be knowledgeable regarding that distinction. In that case I suggest that it would be prudent to understand and have confidence in the quality of the products and service of the original source supplier, before purchasing any chess sets on Amazon. In fact, in my view, the majority of chess sets sold on Amazon are junk. Which is why I prefer to deal directly with suppliers/retailers I know and trust. Thus the adage - caveat emptor.

Ebonizing Wood...
https://www.wwgoa.com/video/ebonize-wood/
How to Ebonize Wood?...
Note that depending on the specifics of the process and substances used to create the stain, the wood grain can be discernable in the finished product...
https://makerdesignlab.com/experiments/how-to-ebonize-wood/
A Few Kind Words About Ebonised Chess Pieces...
https://www.chess.com/forum/view/chess-equipment/a-few-kind-words-about-ebonised-chess-pieces?page=1

Ebonized stain will not chip since it penetrates entirely into the wood. (see the links in my post #15 above.)
Paint may chip, as it resides primarily on the surface of the wood.
Stain is not paint.

@DesperateKingWalk -
And you don't seem to understand the essential distinction between stain and paint. Ebonizing is a process that uses stain, not paint. If some manufacturers use paint and advertise it as ebonized, then they are guilty of false advertising.
I am not disputing anything regarding the wood per se, it's aesthetics or its appeal. I have simply described the difference between ebonizing (staining) and paint.
Enough said. I'm done with explaining this to you.

@Puffin6 You kinda have to take anything Desperate says with a grain of salt since he often just makes stuff up and is very often wrong (just look at some of his threads where he shoots himself in the foot with his own pictures). Here, it's obvious he doesn't understand the process for Ebonized wood. It's not just a marketing term (and not a paint like in the pictures he's trying to pass off as Ebonized wood). RussBell is correct in everything he's said here. It's actually a chemical reaction that darkens the wood itself (not a paint). As such, you can see the wood grain very well (unlike painted wood) and often looks very nice compared to natural Ebony wood. And since the actual wood takes on the chemical reaction into the wood grain, it doesn't chip off like paint. If you're concerned about whether an "Ebonized" chess set is really Ebonized or painted...you can just look at it and tell right away. If it's painted, just complain to the retailer and you should be able to work it out, especially from the major retailers.

@Puffin6 Oh did I mention that he's got issues. But go ahead and read this one retailer's blog and you'll see that Ebonized wood is the way to go (unless you just want to get real Ebony and not use that set). Both woods deteriorate over time...it's just easier to restore the 'stain' than restoring cracked wood, just like that blog says (wait, did he shoot himself in the foot again with his own link to a blog).

@Puffin6 You'll notice that he's debating 'off-topic' and loses that rational thought process by posting a picture of a fairly new chess piece that shows nearly no wear-and-tear and is still covered under warranty. So that would be called a 'defect' in the wood, instead of degradation. Looking at the picture that I posted, the Ebony piece from an old set actually broke in two places, while it's boxwood counterpart from the same set hasn't broken...and this would be a relevant example of how Ebonized boxwood is more practical.
Like I said earlier, you just gotta take what he says with a grain of salt.

@Puffin6 Oh, and he uses a LOT of Photoshop, so just about all of his pictures are fake. Just beware. So when he calls me a 'fraudster', then that's him 'projecting'....oh the irony.

Summary:
One question from the UK OP.
1 answer suggesting indian suppliers
1 answer suggesting a british maker
1 complete answer with links
A lot of discussion and debate about stained ebonized.
OP thanks all the answers.
Old friends meet again and exchange some love. They gave me the opportunity to go back in time and find some nice memories of my fifth grade school time! Thanks guys, you are unbelievable!
@Puffin6
With £150 you can buy very good sets.
If it is your only set, buy a classical weighted staunton design in wood. You are never wrong with one of these.
Don’t forget a nice wooden board too to pair with the pieces.
In the UK:
farrar-tanner.co.uk
Chess&Bridge in London
The Regency chess company (as mentioned by another colleague)
And last but not least, the most famous and expensive
Jaques of London (John Jaques & Sons) but you need to add a zero in your budget.
Good luck with your choice!
Chessbazaar is not good enough when it comes to quality. I bought several chess sets from Chessbazaar in the past and all had nasty issues specially the ones they try to get rid of on sale. In contrast, the quality of Staunton Castle is far superior and their customer service is more honest. As for Royal Chess Mall from my experience their quality is better than Chessbazaar’s but their price is higher and they often get the reproductions wrong because they use images for copying. House of Staunton is another reliable seller with great customer service and free worldwide shipping from my experience.

Lately, I have been partial to World Chess Studio Pieces. A bit more than your listed budget of 150 GBP. They also offer a studio board (folding and non-folding). There are positive and negative reviews, however, I think the quality issues have been sorted out and everyone says they provide great customer service as well.
Regarding designs...
Pretty much everything is derived from Nathanial Cooke's "Staunton" design from 1849 (originally sold by Jaques of London) in some way. I think a traditional Staunton design is a great set to start with... just decide where you want to play and what board size is best and go from there to determine appropriate size.
Knights tend to be the piece that distinguishes a set because of the effort and skill that is required to make it. Some Knight designs are referred to as French Lardy, German and Camaratta. Knight ears may be backwards and fierce or perked up and inquisitive (this can determine the personality of the set).
A popular variation of the Staunton design is referred to as "Dubrovnik". The original version of the Dubrovnik was designed by Pero Pocek and appeared at the IX Chess Olympiad. This is probably one of the most rare and sought after sets as only 50 are known to exist. It's been a favourite by players like Fischer and has been used in a notable Fischer-Spaasky rematch. Andrija Maurovic designed a notable Dubrovnik that is slightly more robust than the original that can make them great if you like to play blitz.
Not surprisingly, there is a history of different Soviet designs. One of my favourite is based on the design of a set used at the 1962 USSR Championship and appears on the cover of Mikhail Tal's book. For this reason, it's often called a Tal set. I like how chunky the base of this set is and how Chuck Grau from this community helped to revive this particular design through extensive research and development.
There are non-Staunton sets but I'm not too familiar with those.

I'm in the UK and have had good experiences with Regency Chess, Chess Baron, Chessgammon, and Librium, all of which are in the UK. The sales are on so you can pick up a set for a decent price at the moment. A classic Staunton style set is a good option, but if you want to stick to your budget and don't mind a simpler design, a set with 'German knight' will be a bit cheaper, and very practical to play with (DGT Timeless set is essentially the same thing).
As for boards, Rechapados Ferrer make unbelievably super-smooth, laminated boards that look great. Some of the big suppliers sell them. But you can get cheaper boards that are still more than adequate. If you'd rather put most of your money into better quality pieces you could always buy a vinyl board in the short term.
Although I'm in the UK, I've also had good experiences buying from House of Staunton in the USA, having bought numerous sets of their higher quality plastic pieces, as well as a wooden Staunton.set. They've always been helpful and happy to replace any pieces with defects. I've never had to pay any import duties or taxes (probably because the value was always below whatever the threshold is). HOS have 30% off at the moment, and free worldwide shipping on orders over $100 (I think). Just another option if you don't see what you want with the UK sellers.
Hello!
I have very little experience in buying chess sets (that is to say that I've never actually bought one!) and so I don't know who are reliable distributors.
I would love some advice as to what websites sell quality goods for reasonable prices as well as recommendations for first wooden sets/ boards (I know a little about designs but would still appreciate guidance if people have sets they adore)
I'm looking for either a set or a board and my price range is up to about £150 (if that helps narrow down recommendations) but could go a little higher if there was something I fell in love with.
Thanks for the help!