How long have you lived and worked in the UK? 20% would normally get you on the ladder and if you have a strong employment history and at least a year left on your visa, it is likely enough now.
Build credit through mobile phone, utilities, council tax, etc. Same ways as in the USA.
Capital One used to have a credit builder credit card when I moved here a decade ago. Low credit limit but no annual fee. And I was able to switch my US AmEx to a UK AmEx. So those may still be options.