I just went through this process. The key thing I did not realize is that UK credit, credit history, debts and earnings are separate from the US. This means I was able to qualify for a US mortgage even without selling my home in the UK. It also meant that having filed my US taxes was key as that proved my earnings. I would suggest making sure all of your taxes are up to date.
Keeping your credit report clean is important and not having new lines of credit, and credit enquires, is helpful. This meant I used my existing UK credit card until I closed on my house instead of opening a new US credit card account. Similarly, I want with a pay monthly no contract cell phone. You may want to hold off on buying/leasing a car and if you really need one, buy a cheap one for cash.
The lender will need to source the funds of your down payment. You will want the down payment in a US bank ideally for 30-90 days or for it to be wired from a UK bank where the money has been for 30-90 days. Do not fly over with a large sum of cash as that will make sourcing the money difficult.
The US housing market moves quickly. Going from offer to closing (completion in UK speak) generally happens in 30-60 days. In compete markets you will need a prequalification letter from the lender to make a competitive offer. You will need cash on hand at the time of the offer.