alex,
I was going to suggest the process in FAA AC 61-143 but it appears this too requires some flying with an FAA instructor. And I found this statement in Table 2. Not sure if it'd apply to you. You may have already read the mentioned AC:
"The UK is no longer a member of the EU as of December 31, 2020.
Therefore, a UK pilot license may not be converted to an FAA pilot
certificate under the TIP-L. UK pilots may still apply for an FAA
private pilot certificate based on a foreign license under § 61.75."
But look through here and maybe you'll find some help. And yes, I didn't read every line of this AC:
https://www.faa.gov/documentLibrary/.../AC_61-143.pdf