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?
There's really no possible problems going just a wee bit smaller. Even going much smaller would probably be fine.
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.
On the size of your current tire there will be the measurements in inches that you are used to, and there will be another measurement that consists of a two-digit number, a dash, and a three-digit number. Both figures are in millimeters.
The most important part is that the larger number must be the same for the replacement tire as it is for the current tire, or else it will not fit on the same wheel.
The smaller number can vary, but not by much. You can generally get away with a few millimeters either way unless you are already using the very smallest or very largest tire your wheel can support.
I don't have any affiliation to this company, I had to buy a 750 grams rim to lace a 1500w motor, and I found that they recommend 1.1- 2.2 wide tyre on a 19mm rim. If they advise nearly more than an inch of flexibility on their rims, the tyre industry doesn't have all different bead standards, and you will have more than .1 inches to play with for the same kinds of rim width:
INNER RIM WIDTH: 19mm
TYRE WIDTH: 28-57mm
https://www.ryde.nl/andra-30 (formerly rigida brand)
Rims for 2 inch tyres are often 12 to 20mm. compare the width with new bike rim specifications.