My UK passport has my maiden name and my green card has my married name can I travel or do I need to change my passport name to match my green card?

You will obviously have problems if both documents are not in the same name. If possible change the passport to match your green card.

It may make it a bit simpler to change the passport name, but it shouldn't be a big issue. I'm travelling to the UK in December, Green Card is in married name, Passport in maiden name... I've been advised to book the plane tickets with my maiden name (the name on the passport) and carry my marriage license just in case I get questioned as to why the names are different.

Buy the ticket with name in passport.

When you return to US, you will need to show your GC to immigration, which in this case, it's recommended to provide a copy of your marriage license. No problem at all!!