If you fly a 'permit to fly' aircraft you dont need to convert.
As per p8 of the following
Pitts_S-1C_G-BTOO_07-23.pdf (publishing.service.gov.uk)
Articles 137, 138 and 150 of the UK Air Navigation Order (ANO) 2016 permit a pilot to fly an aircraft with a ‘permit to fly’ on the privileges of a pilot’s licence issued by an ICAO compliant third country. The pilot held a valid FAA private pilot’s licence with a current single engine land airplane rating and a valid FAA third class medical.