Which bank provide forex trading service?

Fineco bank do, though they're Italy based. I don't use them for share trading etc, however for currency exchange they offer great rates, even better than Transferwise

https://finecobank.com/uk/online/

I've also heard that Fineco bank offered great rates. But much has changed during the pandemic. The terms of my bank became unprofitable for me, so I had to look for another one.  If we talk about trading, I had an unpleasant experience in it and lost money because of fraud. My mistake was that I started trading without knowledge in this area and didn't find a good broker. Actually, I want to give trading a second chance, so I'm learning the basics. And I've found a great blog for beginners, where you can learn what is forex trading and how does it work. Good luck to everyone!

Yes, you should be looking for a broker rather than a bank. There is a huge gap between the most and least reputable broker. There is also a huge difference between the best and worst spreads.

You mention that you have a small budget so you need to be particularly aware that you are likely to be encountering providers of margin trading. You may think that you are only risking £1000 when you deposit this amount but, if things move against you, you will be asked for more funds (the providers prefer to ask for funds rather than closure of positions as they end up making more money). My advice, if you have £1000 that you can afford to lose is to start with £100 or a demo account and try to avoid depositing any more until you have made a profit with the £100. It's normal that, if you have not tried forex trading before, you will lose money while you learn. Hopefully you will only lose a small amount. If/when you find a way to consistently make your £100 grow, you can put more money in. If you cannot make a profit from £100, you should stop and put things down to experience.

You need to be aware that the buzz that you get from winning is much more than the downer you get from losing. It is a lot easier than most people realise to get hooked and to end up losing unreasonable amounts. The notices about "80% of accounts lose money" are genuine, although the figure does differ slightly from broker to broker.

If someone has offered you a certain way to make money from forex trading, (for example, a VIP account at an overseas broker with promises of inside information or trades that consistently make a profit) it's a scam.