Do you have a picture? Are you sure you have a Nelson bubble lamp? We've had lots and while I have huge hands, I've not had a problem getting them through the opening on the bottom to change the bulb.
I agree a picture might help, but if the hole is really too small to fit your hand through you could build a simple tool with a bit of wire and cloth wrapper around the wire. Push it through the hole, unscrew the old bulb and screw in the new one the same way.
