As someone who seems to be genetically incapable of packing light, I tend to go "overweight" every time I fly. Normally this is only a few KG's but I once checked in 80Kg (admittedly when travelling in F) without a murmur or even comment from the check-in staff.
The only time I have ever had a problem (and that was with BD rather than BA at the start of an ex LIN) was when a bag exceeded the 32Kg health and safety limit imposed by BAA at LHR and I had to buy an additional bag at T1.