I can only talk about the basic Honeywell setup.
You have 2 transponders which are selectable on the panel.
If one transponder fails and the tcas goes offline you can select the other one and it comes back online.
If both transponder's fail you won't be visable to other aircraft or ATC the TCAS will go into FAILED mode and neither TA's or RA's or for that matter targets appear on the display.
In my limited experence the failure of a transponder is the least of your worries and easiest to fix for the TCAS dropping offline.
The box of tricks has multiple data inputs from various other instruments and any fault and it drops out. And for any retrofit TCAS installation there seems to be a bit of luck involved trying to trouble shoot it.
Our MEL dispatch with TCAS U/S is 6 flights. Which can be a bit of a swine because we know that the real issue is the rad alt which is a 10 day defered defect but as its part of the data bus for TCAS and EGPWS it takes them with it and then its limited to 6 flights. But this isn't as bad as the transponder failures because other aircraft will be able to get TA''s and RA's.
So as Gulf states the whole proccess of getting a snag fixed can take weeks as the engineers try and trace whats annoying it. They have an annoying habit of working on the ground on the test kit then failing as soon as you get the gear up.