Thanks ksand & larrabee!!
Excellent. That’s what I’ll do. Especially since I just found out our flights for the summer are booked in my maiden name!
(Going through the application online, however, it seems like it might be possible in my married name? It asks if the name on the naturalisation certificate is my current name, and gives the option to submit previous names...then I assume you would send in the passport entered on and the current foreign passport. It seems like the main restriction is that the UK won’t issue a passport in a name that doesn’t match foreign passports except in special circumstances. It would be interesting for someone to try...but I’ll go the safe route! Haha.)
Yeah, I believe the name on your passport MUST match the names on the documentation you provide... which for you will be both your naturalisation certificate and your US passport, which themselves must match each other too (so you can't have the naturalisation certificate in one name and your US passport in the other name).
It used to be that you could chose which name you wanted for your naturalisation, so you could have your US passport in your maiden name but have your citizenship issued in your married name, and then just apply for your UK passport in the married name. However, they changed the rules a couple of years ago and now you can only get your citizenship issued in the name on your US passport and BRP, and therefore you can only get your UK passport issued in that name as well.
The only way to get around that is to change your name in your passport before you apply for ILR, so that both your passport and BRP are issue in your married name, and then you can just apply for citizenship and your UK passport in your married name too.