So I finally decided to bypass the heater core to have AC again, did the usual and it seems that the AC compressor isn't kicking on.

I get air out of the vents but it's not cold, kinda warm.

We recently had that bracket off to do the water pump job, all the connections are connected nice and tight.

What should I be looking at first?

No, I didn't bother to do a search

 

Check pressure in the system. Too low (or high...) and it won't run.