Reversing flow of WAL shipping

Hey Folks,

I want to run a warm standby scenario by you. I'm pretty sure it'll
work, but it's a very large database so even the slightest mistake can
mean a major setback.

Scenario:

Server A is the provider node, shipping WAL files to Server B. Server
B is destined to become the provider node (newer hardware), and has a
sibling Server C that will be the warm standby.

Here's the question: when I turn Server B into the production/provider
node, it's going to switch into the next timeline. If I then attempt
to put Server A (or Server C) into recovery mode, will it switch to
the new current timeline, or complain about being in the older
timeline? Do I have to take another full backup of Server B after
it's become the production/provider node?

Thanks,
David