Whilst it does depend on the circumstances, in my experience, 75% of the time, being 'bumped' occurs at the gate. The most common reason for involuntary denied boarding seems to be oversold cabins, normally the first indication of this to joe public would be a PA announcement asking for volunteers to receive some cash incentive not to travel, following which, they will select someone from the pax manifest -I am not aware how this is done, but (touch wood) have never been bumped off a flight since being Gold, although BA has tried to bump me from First to WTP before on a 12 hour flight from NRT-LHR, to which I told them I would rather go home a day or two later...

Another experience I have had is at check in, if you are cutting it fine (not late though -say 50 mins prior to pushback), they may issue you a BP with no seat allocation on. This can mean that all seats on the plane have been allocated to other passengers, and that you will be bumped if everyone who has a seating allocation appears at the gate.

Other than for a few exceptions, airlines tend not to give notice of bumping people because they don't want to have to do it -overselling flights has been a habit of airlines for years, as it maximizes the revenues and capacities of networks and works fine most of the time. Airlines only panic when everyone for an oversold flight has checked in!!

So, tips for not being bumped:
1. Travel with infants -they will only bump families as a last resort
2. Have some type of special need! -they don't like to bump disabled/special assistance pax.
3. Become a woman and travel alone -this is just a theory, but I would imagine that BA would be unlikely to leave a woman travelling on their own in some outstation.
4. Get FF status -they are unlikely to bump their most important customers.
5. Check in early/online -If you check in early, the risk of not having a seat number on your BP is minimized, although it does not stop the airline from bumping you at the gate.
6. Book connections -if you are a connecting passenger, you are unlikely to be bumped because it causes more trouble for the airline.
7. Check luggage in -they will most likely bump those who are hand luggage only to start with.

I honestly don't know if your fare class makes any difference, I think it more relates to your 'profile' as a passenger.