I thought a brief answer may be helpful here:
My question is whether I can officially accept payment for freelance work
It's 100% normal and commonplace for people of 13 to work and be paid.
(Think of say child actors at one end of the payment scale, and babysitters / car washers at the other end.)
You can and should start tomorrow.
and, if so, what taxes are involved.
You pay taxes exactly like anyone else. There is absolutely on difference.
Note that you will be "self-employed" - there are about four million self-employed people in the UK.
Obviously you will have to read a couple web pages on "how to be self-employed in the UK". But it's exactly the same for you or someone who's twenty.
(You are putting yourself HUGELY ahead of the curve by getting this stuff out of the way before you get older.)
PS don't forget all the computer gear you buy from now on is a business cost.
Also, can I send invoices or am I better informally asking for payment?
Invoices are really "1950-ish" particularly in software. Usually you just send an email these days.
If someone asks us for an "invoice" I just open Pages, type "Invoice: $x,000, date X, to company name". Then I save it as a PDF and email that. It's a non-issue. (Note that you DO NOT need a "letterhead" or anything silly. Completely plain white PDF page.)
Don't forget with software, web sites etc you ONLY get paid in advance.
If you don't have much experience you break it up in to pieces. Example, you're doing the corner shop's web page for £3000. You tell them "Great, it's £250 to do the basic design." They give you £250. You do that work (quickly). They approve etc. the front page. You tell them "Great, it's £1000 to get the site working on aws with your credit card integration". They give you the £1000. You do that work (quickly). And so on.
If they don't pay you in advance for the first step, even if just a small amount, you don't have a client, forget them and move on.
It's old fashioned, but click online (or walk to the corner stationery store) and spend £10 having business cards printed. Just have your name, number, email and "Web and Software Development". You'll be amazed how often you hand one out.
Like, you're Uncle Joe is saying "There's a good lad, making web sites are we! hi ho!" You can say, yes, please take one of my business cards.
It will make you feel solid and organized.