Just to add, if your license needs to be converted to EASA, you need to be careful. Make sure you have all your ducks in a row, in the same row, facing the same direction, standing at attention. And if you're not sure about a certain duck, make sure you contact the CAA beforehand to check.
Due to various issues my conversion took nine months - mostly because the CAA took its time responding to my questions by e-mail: 40 days on average.
First of all, what you need to fill in is form SRG\1104. Read the form carefully. Obviously your base license should not be expired or anything, and you need at least one valid type or class rating. Those are the easy and obvious things.
Your medical needs to be administered by the same *AA as your license. This means that your medical may need to be transferred between countries. This requires a SOLI (State Of License Issue) procedure which can get very lengthy. There is a CAA form but you need to contact the other *AA or your own AME for details.
The CAA needs to have a current (non-expired) LPE (License Proficiency in English) on file, with at least a level 4. You can get this from your friendly local flight examiner, assuming you qualify for a level 6. If you don't qualify for a level 6 you're in a bit of a problem since the CAA is very, very picky wrt. the language institutes that they accept certificates from. I did an official LPE exam at an official LPE language institute in the Netherlands (the CAA examiner route was unfortunately not available to me at the time), got a six, but the CAA would not recognize this. Took me weeks to sort it out.
If your JAA license says "Language Proficiency: English" now, then most likely this refers to the grandfathered level 4 that everybody got when the LPE requirements came into force. That level 4 is expired now. So that text is nowhere near sufficient for an EASA license and technically speaking you may actually be flying illegally already. Somewhere in the last few years a CAA examiner has to have ticked the correct box on a revalidation form, and sent that to the CAA, for your LPE to be registered properly. And since forms do get lost along the way, it's best to phone the CAA to check your LPE status, well before you submit the paperwork for the EASA conversion.
If you think you qualify for additional ratings that were not a rating under JAA (such as aerobatics, mountain flying and a few others), make sure you apply for these ratings at the same time as you apply for the EASA license. It costs nothing now, but it will cost you significantly if you add them later. Various ratings have various grandfathering schemes so it's not that hard to get them. (For aerobatics for instance, submit your BAeA Proficiency Card and you're sorted.)
Upon reception of your shiny new license, check it carefully. When they issued mine they conveniently forgot to include my aerobatics rating, despite me adding a cover letter that I was applying for one, and despite ticking the correct box on SRG\1104.
Rumor has it that the CAA counter service at Gatwick is very, very much improved. If you have all your ducks in a row, then you can walk out the door with your new license inside two hours. But you have to have all your ducks in a row. If you arrive at their counter without some sort of valid proof of LPE 6, for instance, they will not and cannot help you.