The suggestion was slowing down to allow the service to be completed - on SH staying airborne for another 10 mins is often an increase of 1/6th of the total flight time. I do not believe that any reduction in cost index (by the way a concept that any modern operator is aware of not just BA) will mean that staying airborne equals a lower fuel burn. Ground idle fuel flow on the engines means even if the engines are running for the same amount of time you are always better off on the ground - that's without shutting one down.
Juan,
On a typical shorthaul sector (90 to 120 minutes), I can reduce the CI and arrive perhaps five minutes or so later having burned the same amount of fuel as flying at a higher cost index and arriving earlier. If we still have to wait for the same stand, this means that the amount of time spent waiting on the ground with the engines running is five minutes less. Ergo, we use less fuel overall.
Slowing down is not a big issue in terms of cost and in some cases is a benefit. In extremis on a longhaul sector, you could slow down enough to arrive an hour late whilst reducing your fuel burn by 10%. So why don't we fly slower all the time? In the long run, you complete fewer sectors for a given aeroplane, which reduces the overall revenue.
However, if we're significantly early on a particular sector, slowing down to allow the service to be completed whilst still allowing us to make the schedule, has a lot of benefits. The main one for you being that BA won't b*ll*ck the SCCM for not completing the service. If in doubt, ask.