Get your ICAO license upgraded to ATPL before you do the conversion.
To convert an ATPL, you need to do all (14) exams and a checkride - on the aircraft you are rated on, and this can be done in the sim.
You need to meet all requirements for ATPL issue - check LASORS on the UK CAA website.
Brookfield Aviation might be able to help - at a cost. Not sure what the current "deal" is but you might be able to work on a 1-year validation, which will give you time to meet all the other requirements.
Good luck