Yes, all this is fine.  Some people have been denied entry to the US when coming to the US *temporarily* to marry a US citizen, so you will see advice online (even sometimes from immigration lawyers) saying that you need a fiancee visa for this purpose.  But the State Department's Foreign Affairs Manual says that [a B-2 visa is appropriate for this purpose](https://fam.state.gov/FAM/09FAM/09FAM040202.html):

> 9 FAM 402.2-4(B)(1)  (U) Fiancé(e) of U.S. Citizens or Permanent Resident Aliens

> (CT:VISA-778;   05-13-2019)

> (U) An alien proceeding to the United States to marry a U.S. citizen petitioner within 90 days of admission is classifiable as a K-1 nonimmigrant under INA 101(a)(15)(K).  (See 22 CFR 41.81.)  The fiancé(e) of a U.S. citizen or lawful permanent resident (LPR) may, however, be classified as a B-2 visitor if you are satisfied that the fiancé(e) intends to return to a residence abroad soon after the marriage.  A B-2 visa may also be issued to an alien coming to the United States:

>- (U) Simply to meet the family of his or her fiancé;

>- (U) To become engaged;

>- (U) To make arrangements for the wedding; or

>- (U) To renew a relationship with the prospective spouse.

To overcome a possible skeptical immigration officer at the border, you can bring evidence of your intention to return to the UK, as well as evidence of your fiancee's plans to move there after marrying you.  But I would not offer this evidence before it becomes apparent that the immigration officer is questioning your intention to leave the US.