This has been covered many, many times over but for what it's worth:
The flying you do depends on the fleet you are recruited for. At the moment, the company is recruiting for LGW which is a mixture of short haul and long routes on the 737 and 777.
Long haul routes are:
Tampa, Orlando, Altanta, Kingston, Bermuda (shuttle to Port of Spain), Barbados and Antigua (shuttle to St.Lucia, Grenada and Tobago). Houston and Dallas will have moved up to LHR before the next batch of new entrants are trained.
Short haul trips will vary from day trips of 2-4 sectors to 3 day multi sector trips. Please correct me if I have this wrong.
You have to wait for two years before putting your name on a transfer list. After that, it could be a 3-5 year wait (5-7 years in total) before actually achieving the transfer, depending on your seniority.
With hundreds of people on transfer lists waiting to get to long haul at LHR, it is unlikely that they will recruit directly from outside, at least on a permanent basis. In the last two years, they have recruited temporary crew for long haul in 3 different campaigns. Most of these crew ended up at LGW. None of them remained at LHR on long haul.
It's important to note that BA will not recruit anyone directly into a particular fleet or rank just because a candidate has prior experience. There are very strict transfer and promotions protocols that are part of the terms and conditions agreed with the relevant unions.