You will also need to convert to a Part-FCL licence.
This means you need English Proficiency checked (which could be done by an examiner at the same time as renewing your SEP rating) which is often a sumbling block for people.
You apply for this online:
See here:
Converting UK and JAA Licences and Ratings to Part FCL | Private Pilots | Personal Licences and Training
Online application form here:
Apply for a Licence Conversion | Private Pilots | Personal Licences and Training
However might be easier to stick with paper if also renewing the class rating at the same time.
As mentioned, details to renew your class rating are here:
Renew a SEP Aeroplane Class Rating | Private Pilots | Personal Licences and Training
For the conversion, there are many other threads on here...
Your US medical / BFR mean nothing as far as your UK licence is concerned.