Hi Solaire
Whether or not you need to file a UK tax return will depend on whether or not you are considered self-employed.
If you are considered doing casual earnings and it's under £2500, you may not need to file a tax return but you should still contact HMRC for clarification and let them know. HMRC are very approachable. I believe there is even a web chat however I have never used it. You can even register for a Personal Tax Account
I assume that your status at the moment is one of only a US citizen however if you ever want to take up British citizenship in future, you will be assessed if you are of “good character” and tax issues will be taken into account and can affect your application so it pays now to start doing the right things and keep in communication with HMRC about what you are earning in untaxed income even if no tax is due.
Regarding the US side of things, the important thing to note is that there is a Foreign Earned Income Exclusion (FEIE) for earned income (wages) overseas and the threshold is in the $100,000 region. So you will definitely not owe any US tax based on what you said however you still need to file if you are over the filing threshold and claim the FEIE.
Again you should read up on publication 54.
Forms you might need to file are (I would think the first 4 might apply now)
Form 1040
Schedule B
Form 2555 (this is for the FEIE)
Form 8965 (this is the health coverage exemption)
Form 8833 (possibly if you are ever part of a work pension scheme)
and possibly form 1116 for foreign tax credits if needed in future