DERG,
First off, the atmosphere does not always follow a linear path in temperature.
I have seen it quite a couple times where climbing actually increases the SAT (static air temp). A lot of the time it has to do with troposphere/pause/stratosphere.
Secondly, yes increasing the oil temperature helps warming the fuel however flying faster increases the TAT (total air temperature) due to more skin friction. This increase of temperature can be in the order of 4/5C. Depending on how much you can accelerate of course.
Thirdly, the number of engines is completely irrelevant.
Fourth, we do not get instructions from BA Ops to ''save fuel''.
Best practice dictates that if fuel temperature does not look like it's going to be a problem then why not climb if it saves fuel?
Plus we are largely second guessing to why BA038 climbed. It could well have been that their Mach No was restricted on the few airways that connect Europe to the Far East.
Hence climbing and increasing the Mach No might have been more beneficial then sitting ''low and slow''.
At any rate it's complete guesswork.
As long as the low fuel temp warning (3deg typically of Fuel Freeze Temp) does not illuminate and the forward flight plan has been asessed for OAT's then climbing ''should'' have absolutely no impact whatsoever.