I do the Forest Hill to Canada Water commute every day and although busy the people using the trains to London Bridge takes the pressure off. The overground trains are frequent, clean and not overly crowded considering a comparison to other lines. Forest Hill is far enough done the line so that you can get a standing place in the middle of the seating area to avoid the squash by the doors. Also there is always the option of the trains from catford that is particularly useful for me by using Charing Cross for West End outings. Overall FH is very well connected.
As you plan to travel to London Bridge you will have the luxury of choosing either the overground or the train you may get lucky on occasions and get a seat on the train. Before the overground arrived it was a bit hit or miss if you could actually get onto a train at peak times. It is much more civilised now and I have not had any problems.