In BA, only the A320 series aircraft are equipped for MLS. LHR provides MLS approaches to all four runways. BA expect all suitably equipped aircraft to fly MLS approaches where they are available. On the current network this is only LHR.
The reason for this is, MLS allows closer approach spacing (and therefore, flow rate) than ILS in LVPs. All down to the same Cat 3b minima of 75m, no DH.
ATC weren't able to utilise this improved flow rate, as they said they didn't know which aircraft were eligible for reduced spacing. So in order to 'help', BA expect their SIAs to always 'fly the wave'! This was an attempt to satisfy ATC's concern and enable the higher flow rate in LVPs. Sadly this has not happened. Possibly because a few A320 have never been suitably equipped.
So in short, apart from being an additional 'back up', and the fact that microwave is a more solid and reliable signal than ILS, there is absolutely no benefit whatsoever. Just a huge amount of significant expense. I make no comment as to whether this expense, is the only REAL reason BA A320s fly the microwave at LHR.