If you are spending all of your time in the UK and you are only earning income in the UK, you are likely a UK resident for the bank questionnaire. However, you are a U.S. citizen, so they will report your UK bank account to the U.S. IRS. The bank may ask you to complete a W-9, or for your U.S. social security number. This is so they can report to the IRS.
You probably don't need to worry about FEIE residency, as long as you spend at least 330 days outside the U.S. during the year (or any 12 month period). You probably meet the requirements to exclude your earned income. You should look at Form 2555 and IRS Publication 54. You might be better off claiming foreign tax credits in the long run, but the exclusion is simpler.
State residency can be tricky. It depends which state you are from and other factors. You may still be resident in the state you left. The state residency rules are usually included in the instructions to the state nonresident tax form.
I hope this helps.