This is going back a bit, 1996, but I got a secured credit card with HSBC. I showed them my passport with the stamp that indicated I was in the UK as a spouse, my marriage certificate and my lease agreement with my name as a resident and a £500 deposit. I had only been in the country six months (I originally arrived on a fiancée visa). Once the year passed, it became "unsecured" with a credit line of £1500 and my deposit was returned. Perhaps you can try that? It may be different now but it's worth a try.
I did have a bit of trouble getting a dumb catalog because I wasn't on the electoral roll but I ended up sending a copy of my passport visa entry stamp as a spouse and my lease agreement with my name. It was ridiculously time consuming for a blinking Next catalog but I didn't have much trouble after that. Again, go visit the banks and find out what you can do to get credit.
I've just noticed I have had that HSBC credit card 20 years.