Peter Goodland wrote:

At London Bridge, stay on the same platform and get the next train to
Waterloo East.

That applies ONLY if you take a First Capital Connect (formerly Thameslink)
service. If you take a Southern service from Gatwick you will end up in the
bay platforms - however, it's still a level transfer via the barrier line
and the subway (which is ramped) to platform 6, where the Waterloo
East/Charing Cross services leave from.