If you haven’t heard of this before there’s a good introduction to the project at https://gds.blog.gov.uk/2015/07/23/making-payments-more-conv...
There is actually a legal requirement for banks to offer no-credit no-fee accounts, in order to help people get into the banking system and take advantage of the associated discounts, but obviously they don't advertise it much.
Proof of address used to be easy as any utility bill would do, but now days it's trickier because so much is done online and paper bills don't get sent out much any more! And, of course, it's a problem if the bills at your house are in someone else's name. (They won't accept print outs from the Internet for obvious reasons...)
However, if you're registered for paying tax in the UK then you would certainly have a letter from Jobcentre Plus or HMRC showing your address. This would be accepted by your bank.
Another option is to get a UK driver's license, which is pretty easy and inexpensive if you already have a driver's license from your (EU) home country.
I had a similar issue when I was opening a bank account a while ago. I didn't actually get any bills in the mail, so I printed off my ISP bill, which was a pdf of what they send you in the mail. Obviously, the printer didn't fold it to fit in an envelope.
The person at the bank asked if it was a printout or actually from the mail since it didn't have any creases in it, they said they could only accept it if it was from the mail.
I told them I would just walk outside, fold it up, and walk back in, it was exactly the same as what you'd get in the mail. They ended up accepting it, even though technically they weren't meant to.
It's ridiculous, I don't actually physically receive mail from anything. All my bills etc. are done electronically, I even sign contracts electronically.