Why don’t I have power to new light fixtures I installed?
The thing about GFCI

GFCI isn't a special kind of socket.  It's a zone of protection.   Inside that zone, the current going in and out is monitored.  (electricity flows in loops, that's why every plug has at least 2 prongs).  If the 2 sides of the loop are equal, all is well. If not, current is leaking and could be shocking someone.  

Here's an example of a zone.  Some GFCIs are circuit breakers.  The GFCI breaker places the entire circuit in its zone of protection.    

You're well familiar with the GFCI receptacle.  Obviously, the GFCI socket places the sockets into its zone of protection, I mean duh :)  But actually, on the back of the GFCI receptacle, covered up with a warning tape that says (very roughly) "For Wizards Only", are two special terminals marked LOAD.  They have one purpose: extending the zone of protection to more of the circuit. (And they should never be used, ever, except for that one purpose - that's why the LINE screws take two wires).  

So even a GFCI socket somewhere else can protect other stuff - lights, fans, and other sockets.  

Think about it:  That socket in the bathroom must get its power from somewhere, right?  Very likely that place is itself some sort of GFCI device.   That would explain why the bathroom doesn't have a GFCI receptacle if it's already under the protection of a GFCI device somewhere else, it wouldn't need one!  

Anyway... I have a sneaking feeling that you changed these lights with the power turned on.  And I have a feeling you touched a neutral wire to ground, or did something to cause imbalanced current to flow, and this tripped a GFCI somewhere.  

So you need to go hunting for that GFCI recep, breaker or deadfront that has tripped.