If you don't have 500 hours on a multi pilot aircraft (such as a Dash 8, 737 etc) then you'll need to do a full multi pilot endorsement CAA observed LST (lic skills test) to get your JAA ATPL. (eg, JAR 25)
If you have 500 hours on a (eg) 737, either seat, then you just (hahah, just!) need to brush up on what the CAA require and do a CAA observed LPC (Lic prof check) on the 737. If you have 500 hours or more on a Dash 8 but want a 737 rating, then you'll need to do the full JAA 737 rating. etc.
None of the above are cheap....
If you're currently flying a Bandit or Metro (JAR 23) in 2 crew operations then you'll need a letter stating that from your employer and under what regulation you're flying.Eg RPT etc. If you have 500 hours or more multi crew, then the top one applies.
Regarding the airline assisting in your conversion process, I doubt it unless you had exceptional skills that they could use.
However, this is all inclusive of having at least 1500 hours or more. (which you do). I am assuming you have either a visa or a passport.
Jobs? There's plenty at the moment for experienced crews.
There's a lot of biz jet movements too so they're worth a look-in and most don't charge you for the endorsement.