Automatic switching to battery on mains failure?

The MAX1555 is what I use for Li-ion. The Li-ion works well for a 3.3V
system. You get 300-400mV of voltage for a LDO.
Pictures of myt MAX1555 based charged are at Loading...
I did not add powerpath control to the board. I may in the future.

The device you linked to is a NIMH charger. That should work for charging too.
You could use either charger with an LT4412 (or other powerpath controller).

I believe if you check the Linear Technology, TI and Maxim sites you could
probably find a single IC that will do both functions. Chances are the
part will only be available in SMD packaging.

(* jcl *)