Seriously, do both. It just makes total sense and will give you the opportunity of applying for more jobs than most.
The cost isn't much higher than doing UK CAA alone as well.
Your UK PPL is fine to start the EASA and UK ATPL exams. It's also fine to do the EASA and UK CPL. There's no need to convert anything.
The first thing you should be doing is the class one medical. I hope you already have a UK Class one so you'll only need an EASA one. There are several places in the UK where you can do an initial, though it will be cheaper to fly off somewhere like Romania to do the initial there which some people prefer. But it'd much easier for you to have the dates aligned so I would suggest revalidating your UK one at the same time as the EASA initial even if the UK has a fair bit of time left on it.