What do pilots landing on aircraft carriers do when the ship is not visible due to weather?
My experience is circa 1990s, but I can offer some perspective on US fixed wing operations.
Besides TACAN and ASR for non-precision approaches, there are (were) 3 precision instrument approach options available:  Precision Approach Radar (PAR), Instrument Landing System (ILS, or “Bullseye”) Automatic Carrier Landing System (ACLS).
PAR: This consists of both azimuth and glide slope radar that allows the controller to issue verbal advisories to “talk down” the pilot.  This is generally a back up for when the other systems might not be working.
ILS: Although it works on different frequencies than civilian ILS, functionally it is identical.  A passive signal is received by the approaching aircraft which powers azimuth and glide slope indicators in the cockpit.
ACLS: This is an active system that locks onto the aircraft and sends a discrete signal to drive the indicators.  For a pilot flying a Mode 2 approach the indications are identical to the ILS, but there is greater precision as well as the option to upgrade to a Mode 1.  In a Mode 1 ACLS approach the pilot can couple the autopilot to the signal to allow it to fly the approach hands-off.
All these options are similar to what civil airfields have available, with one notable exception:  The aircraft carrier has Landing Signal Officers (LSOs) on station.  These hardy souls man the flight deck during all recovery operations, in all kinds of weather, and are the final piece in helping a pilot get aboard when the weather is bad.  In very poor visibility LSOs can see the landing light before the pilot can see the carrier and can help talk them down on short final. (There are lots videos on YouTube of this...)
Because of the unique functionality of the Pilot/LSO team, the concept of “minimums” and “missed approach” don’t really apply at the ship.  At 3/4 mile from touchdown the pilot will be asked by the controller to “call the ball” in reference to seeing the glide slope indicator of the ship’s Optical Landing System.  The pilot will either call the ball in sight, or declare “Clara” which means he/she does not have the ball (ship) in sight.  On the call of Clara the LSO will take over issuing power and line up calls over the radio until touchdown, or instruct the pilot to wave off if a safe landing is not possible.
And finally, if the airplane is bingo fuel they would be expected to divert unless the ship is operating blue water, in which case they would go to the tanker.