Mandolins are smaller than violins so with BA you should have absolutely no problem as unlike many other airlines they actually let you take it in addition to normal hand luggage. See here: http://www.britishairways.com/travel.../public/en_gb?
p_search_text=violin

Make sure that it is in a hard case, don't make a fuss about it at check-in. Ideally keep it covered with a coat or something, otherwise you run the risk of some clueless agent wanting you to check it. I can get my violin case (which is off the top of my head about 25inches long, 8" wide and 4 deep) into the overhead locker no problems. The main thing is really the case, as you want something which gives as much protection as possible, fortunately I've got the top end model of one of these:
http://www.musafia.com/editorial5.html so can hopefully cope with whatever BA want to throw at me!

The other factor to take into account is that you MUST keep the instrument well humidified in the dry cabin air. If your case doesn't have a humidifier in it, then my recommendation is that you get one of these small plastic soap holders or similar, drill some holes into the top, then cut a piece of sponge to fit the container - soak the sponge in water - wring out so no drips, then put into the container and you have a very basic but serviceable humidifier which will hopefully stop any sudden cracks from forming due to the sudden drop in humidity. You should check first with the owner, but if it is a particularly delicate instrument, you may need to very slightly loosen the strings too...

Incidentally, HighLife, is your brother well known? I might have worked with him... Buying tickets for celli is one of the most frustrating aspects of looking after cellists.