Yep, i go from the north side of Bexleyheath everyday.
You have a few options, but the quickest and easiest route is via plumstead, woolwich, greenwich and then carry on Evelyn Street and Jamaica Road to either Tower Bridge or London Bridge. It will take about an hour. Most of it is on busy road with cycle lane or bus lane. It is busy, fast roads so not very relaxing. Basically follow A206.
I have a quieter route via Plumstead Common, Woolwich common, and Greenwich Park, but that may be a bit out of your way.
The one road to avoid is the dual carriage way through Thamesmead.
In the summer you have a great slow route via the Thames path, which is fantastic.
Let me know if you have any other questions.