I never get the opportunity to talk about an old pet project of mine, so when I can... here comes something longwinded and bo-o-o-o-ring;

BA has priority baggage services, however there are differences due, largely, to technology. Brief excursus: baggage is segmented by commodity codes, assigned to each bag in Flight Management (aka the load-and-balance cousin of Altea CM). Basically, on long haul

• Transfer passengers will have a BT commodity code. Work was underway to divide between BTS and BTL (less than 90 and more than 90 minutes' connection), but if I remember correctly there were issues with another system that allocates the baggage bins to drivers.
• Premium passengers will have a BP commodity code, meaning that they will be in a Premium bin.
• First class passengers will have a BF commodity code, meaning that they will be in a First class bin.

The theory is that the BF will be out by the door, followed by the short-connecting bins, then Premium, then everyone else. The commodity code helps the system allocate baggage bins and, in those stations where the baggage operation is guided by a Baggage Reconciliation System or BRS, it will prevent a baggage handler from, say, loading a transfer bag into a direct bin. The loader 'zaps' the bin code and the bag code; if the bag is not meant to go in that bin, the gun will display an X, vibrate, send electric shocks, call mommy and so on.
Now, there's a surprisingly large number of airports that don't have, or use, a BRS. The reasons are plentiful but they can be summarised as costs and complexity. A seasonal route doesn't need a BRS and a route with less than 50,000 bags/year doesn't either. In those situations specific bins will be created by FM, and then the handlers will manually ensure the bags go in the right bins. And they normally get it right.
The other thing to bear in mind is that this is a manual process, and as always humans are the weak link - especially if humans happen to work at Heathrow. Outstations are normally pretty good in loading aircrafts as per plan, but LHR has a knack for a) not following the plan and b) offloading bins in the stupidest possible ways. It doesn't help that, in LHR, baggage drivers are told by the system which bins to pick up instead of getting whatever is needed to get off first.

As for short haul, only BT and normal bins are created, for very simple reasons of space onboard and consistency. On A319s it's impossible to have BP's due to space (only 4 bin spaces available).