Ive only ever flown my bike with BA (about 7 times now), every time has been hastle free and never asked to pay extra for the bike.
You need a bike box / bike bag (which you can hire from most bike shops), you pack your bike by removing pedals, seat post and wheels and loseningthe handlebars and twisting them horizonatal to the frame. You can search the internet/youtube for more detailed bike packing instructions.
BA let you take your bike as one extra free pieve of luggage (so you can still take a suitcase and hand luggage), the bike and case must be 23kg or less.