BA want to standardise the SH fleet to Airbus. As stated above most of the aging 737 fleet is already on its way to Victorville.
The current meat grinder that is SH is full of pilots wanting to make the move from SH to LH, an aspiration that is central to their giving continued support to the BACC during productivity changes that seem to be pointed squarely at SH. Any change to this aspiration attributed to DEP onto LH as has happened in the past would be met with a huge amount of resistance, so much so that I believe the BACC would have to block any such recruitment plans for the protection of its current members.
DEP onto the LH fleets will not happen for some considerable time, if ever IMHO. BA requires A380 pilots to have BA airbus experience, the 767 is winding down as is the 744, the 787 is becoming the most junior LH fleet with a massive queue of SH pilots waiting for it and the 777 fleet becomes more senior by the day as the 744 guys transfer across.
Whilst I think that DEP will be opened for non type rated pilots again they will all be going to the French Electric SH death jet either at Heathrow or at Gatwick under whatever rostering system exists at the time.