Here's a simplified answer to a complex question.
There are over 3000 pilots in BA with the newest (or most junior) at seniority number 3264. The most senior is, of course, at number 1 and we are all in sequence between. My own number is in the low 600's having worked here for 18 years. Each year the seniority numbers are recalculated totake into account those who have retired or left - I expect to move about 180 places next year putting me just below seniority number 500.
So new pilots joining BA start at the end of the list on the fleet that they are allocated to at the time. This depends on which fleet needs to recruit and whether the pilot is a TEP (cadet) or DEP (experienced). Cadets would normally be posted to the RHS of either the Airbus 319/320 or the 737 - usually at LGW. A DEP with heavy jet experience would go to RHS on 777 or 747.
Once in place, a new pilot is frozen in that position for 5 years. He can still bid to change seats or types each year but is likely to be ignored. Once the freeze is over a bid for a change might be achieved but flying hours need to be over 3500 for command and the longhaul commands tend to be very senior so patience is needed.
There are no restrictions about short to long haul or vice versa but there are experience level restrictions which tend to mean a first move from co-pilot to Captain would be on a shorthaul type....