(Pedantic Brit here) I think most Brits would say 'with a spade' if asked how to dig a hole and would use a hoe to break up soil, disrupt weeds, harvest roots, etc [0].
As a Brit, I would say the same. However it may be some specialised terminology - for instance the wooden beams that support the coffin over the grave before it is lowered are "putlocks", a word I had not come across before. So it would not surprise me if a sexton referred to a spade as a hoe.
(going down a rabbit hole now. I think that's the right phrase)
There are loads of sites that offer grave digging services or equipment (e.g. [0]) although none mention hoes that I can see. However, I did find more about the use of hoes to actually dig holes (coming back to the original subject): [1]. How could I have forgotten about 'mattocks'.
It's possible that the word "hoe" is rarely written down in this context. I wasn't able to find "putlock" in this context on the web, although I found a blockchain protocol where the name clearly linked the words "grave" and "putlock"
In Britain we just call them 'diggers'. JCB is a very common British digger brand though so they do get called that as well. We do have backhoe loaders they are machines with a wide flat shovel for scooping material up and moving it around or loading it into a dumper truck. You wouldn't dig a hole with one but most people would look at it and call it a digger.
