Look at what clearance between the tire and the frame do you have now. Often the tightest place is the bottom stays.
Changing 2.0" to 2.1" increases the tire diameter by about 2.5mm ("about", because the rim width stays the same and does not proportionally increase), i.e. the tire grows ~2.5mm upwards and ~1.3mm on each side. Not all tires are measured equally, some "2.1" tires may be wider than others, so add one millimeter more to be on the safe side. Besides that, knobs make the tire higher, attempt to account for that too.
If you conclude that there is enough space to fit the wider and knobbier tire, there are no other obstacles to replacement.
(Not directly related to the question, but I'd strongly recommend buying higher-quality studded tires. This is an item which brings **much** more value for the money than any other bike part.)