My current bike tyres are 26 x 2.0 (Manufacturer fitted tyres)
Is it possible to fit 26 x 2.1 or 26 x 1.95 on these wheels or is this going to cause an issue?
Slightly smaller is fine
There's really no possible problems going just a wee bit smaller. Even going much smaller would probably be fine.
Slightly larger is probably ok
With the original 26 x 2.0 tires on the bike, check the clearances. Remember that a 2.1 tire will be both wider and "taller". Look at how much space there is around the tire at the fork crown, where the tire is close to the downtube, where the tire passes between the seat stays, where the tire passes between the chainstays, where the tire gets closest to the seat tube... As long as there's room for a wider tire, it'll be fine.
It's really common to change out tires for slightly different sized ones.