I'm unfamiliar with UK banking. Can you not obtain a credit card from a different bank than the one you do most of your banking at?
You can. You can get any account with anyone, as long as your credit check comes up good (simplifying. I'm sure there'll be some type of accounts that are difficult to get).
I wouldn't be too bothered if my bank doesn't support apple pay (would be mildly annoyed) because I use my credit card for pretty much everything. My credit card is with a different company than my bank accounts. If my credit card doesn't support it, then I'll probably switch that (Barclaycard.. But I'm sure they'll support it, because they already do in America and they're big on contactless already - pay tag, a sticker for the back of your phone, and pay band, a contactless wristband for making payments)