What ILS type approach my flights landing could be?
The 3 METARs all indicate extremely poor visibility. The first one (02:30Z) said:

0000 R27/0075 FG VV///

which means visibility less than 50m (0000), Rwy 27 RVR 75m (R27/0075), fog (FG) and vertical visibility not measured (VV///).
The second METAR (03:00Z) still reports visibility less than 50m, but RVR has improved to 100m. The Airbus A321 is certified for CatIIIB down to RVR of 75m (Airbus getting to grips with CAT II / CAT III operations), so technically an approach would have been allowed before, but note that the METAR is only updated every 30 minutes. The pilots were probably informed by the tower or approach controller about the actual current RVR value.
After landing, the third METAR (03:30Z) already shows improved visibility of 100m and an RVR of 750m, indicating that the fog has started to clear.

Here are the minima for the Cat II/III ILS Rwy 27 approach at VILK (Lucknow):

The Cat II is out of the question with a required RVR of 300m (or 350m without autoland in Cat D aircraft). The Cat IIIA would have required at least 175m RVR.
Since the RVR was reported as 100m before landing, the pilots must have prepared for a Cat IIIB ILS approach with autoland. Once the pilots verified that the requirements for a Cat IIIB approach are fulfilled, it does not really matter if the RVR gets a bit better by the time they land, so even if it was 200m at the time of landing, the approach would have been performed with Cat IIIB.

Landings with Cat IIIB seem to be quite common in India according to bangaloreaviation.com:

On the domestic side, only the A320 operators Air India (NACIL-I), GoAir, IndiGo, and Kingfisher have most of their pilots Cat-IIIB certified. Boeing 737 operators Jet Airways, JetLite, and SpiceJet have their pilots certified to the lower Cat-IIIA standard (175 metres runway visibility).
[...] not surprising since the Airbus A320/A319/A321 comes with Cat-IIIB capability built-in, [...]