I am near completion on my first ever attempt at laying tile. I am using
12
x 12 ceramic glazed tile on my bathroom floor and am at a stand still on
attempting to cut a holed for the toilet flange

This isnt how HOYLE does it but I 've cut various shapes in regular and
porcelain floor tile using an angle grinder with a 4" dry diamond blade in
it. The blades are relatively cheap, about $ 10.00 at HomeDEE. I usually
brace the tile but use the angle grinder "freehand" like a Dremel tool (I
have very large hands). You could start by cutting a square piece out then
the diamond blade will definitely "round out" the remainder. Works well for
me.
R