I moved here from the US about 7.5 years ago and went through similar pain to what you are about to⌠I can tell you one thing - your US credit score means nothing here sadly. We had perfect US credit, owned a house and everything, and when we got here DFS wouldnât even let us finance a cheap sofa! 
Itâll take time to build up your UK credit, but basically get yourself on the electoral roll (didnât apply to me as I canât vote here), get a mobile phone plan, then when eligible get a credit card. Once youâve used that consistently for a bit, take out a small loan with a high street bank - the interest rate will be high, but the key is having the loan and paying it on time. Similarly, this runs counter-intuitive to everything youâve been told about managing money, but choose the monthly option on stuff like car insurance, tv tax, road tax, etc. Yes you will absolutely be paying more in the long run, but see it as an expense you are paying in order to build credit quickly. We got phones right away when we moved here, financed road tax and tv tax, got a UK credit card about a year after (Natwest Rewards), then took out a loan with Natwest shortly afterward just to show payment history. Weâve now got solid credit and no issues financing what we need, but it did take time.
Some things Iâve learned: Capital One here isnât like it is in the States. In the US it is a regular bank, well-respected and used by many (it is actually our primary US bank). In the UK Capital One is seen as more of a âcredit builderâ bank, often used by people with bad credit to rebuild. HSBC will be easier to open a UK account if you already have an HSBC US account, but donât seem to be any more helpful if you donât. Use Wise to move money over, never send wires from your US bank as the fees are high and the exchange rate they use is universally terrible. Weâve probably saved multiple thousands of pounds over the past 7 years by moving money over with Wise (my salary is paid in USD to a US bank so I have to send big chunks of money over every monthâŚ) Chase is relatively new to the UK, and seems to be operating more like a fintech over here rather than the very established brick-and-mortar legacy bank they are in the States.
All that said, we started our UK life with Wells Fargo in the states and Natwest here, but now weâve moved our US accounts entirely to Capital One (better HYSA, credit card with no foreign currency conversion fees, easier to manage from here). Natwest is still our âprimaryâ bank here (it is where I send my money each month at least), but we are slowly moving our daily banking over to Monzo. Monzo will do extra checks if you are a US citizen - it took me a couple weeks to actually open my Monzo account because of them - but once youâre in there doesnât seem to be any extra hassle.
Iâm sure thereâs more but thatâs all I can think of at the momentâŚ