Why is taxiing super slow after an ILS CATIIIB landing in almost zero visibility?
The reason for using taxiway C is described in the Low Visibility Procedures from the AIP:

4.4.4 At present, TWY ‘C’ and Stand no 11 to 14 are equipped with CAT IIIB lighting system as per short term interim plansubmitted to DGCA by AAI and agreed to by DGCA vide letter no. AV.20015/09/08-AL dated 15-07-2014. Therefore, duringCAT IIIA and CAT IIIB operations, arrivals and departures shall use TWY ‘C’ only.

So the reason you had to taxi via C is because C is the only taxiway with sufficient lighting to support CAT III operations. For CAT III operations, taxiway centreline lights are required. The other taxiways at Lucknow only have taxiway edge lights.
Source: https://aim-india.aai.aero/eaip-v2/eAIP/EC-AD-2.1VILK-en-GB.pdf
The reason you had to backtrack (turn around on the runway and taxi back in the opposite direction) must be that the plane was not slow enough to vacate via C when you passed it the first time. I know you said you had already slowed down, but in thick fog it can be pretty hard to estimate speed, especially when looking out a side window. The pilots obviously have access to precise instruments, and must have determined that the aircraft had not slowed down enough.