Surname change on passport after filing I-751

To first give a timeline and an idea of my situation:

I first moved to the USA from England on a fiance visa in June 2018. I married my fiance a month later. Life happened, our marriage didn't work out and we got divorced a year and a half later in December 2019. We've since both moved on and got re-married. However, I'm still on my conditional green card. I sent off my I-751 Petition to Remove the Conditions of Residence form sometime this year, I think (can't remember exactly when) and recently received a letter requesting more evidence.
My current husband and I have already been considering moving (back, for me) to England. However, I want to change my surname to his (I kept my own surname with my first marriage) I haven't done it yet, because I don't know how or if it will complicate everything. Is it just a matter of notifying social security, or whoever it is - (yes, I'm completely clueless) or is there a lot more to it?

I know this might be quite an unusual situation, but I'm hoping someone else has a similar experience or might know the answer/s

Thank you in advance.

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