Building credit score on a blank slate (moved to UK in March)?

So I moved to the UK in March to live with my partner. We plan to buy the house we are living in by the start of next year which is our deadline. I just started a fulltime job in May.

My name is on the electricity/gas bils. I've registered to vote but my next step is getting a credit card. Halifax (my bank) won't give me one as I'm too much of a clean slate for them. Are there any other credit card companies you'd recommend that would be more leniant?

Is there anything else I might be overlooking or should consider also doing?

Thanks!