I could not get any cardboard big enough to make a template so in the end I used old computer paper. I did the floor in three sections: the L-shaped bit with the toilet, then the long bit with the radiator pipes, with the door at the other end. After a week of measuring, measuring, measuring again and laying the template down and measuring, measuring some more, I applied it to the vinyl and (gulp) cut it out.
I laid the vinyl on the floor this Tuesday, and while the edges matched to the walls, there were bulges in the middle of the vinyl. So I spent time cutting away from the walls in order to flatten the bulges.
Result: a perfectly flat floor, but with some gaps at the edges of the walls and shower enclosure of as much as 4 mm. Never mind, eh? I fixed it all down with double-sided tape and it looks great. Next task: applying white silicone mastic all round to cover the gaps where the vinyl joins the wall.
The Medway Handyman was right about not doing this for a paying customer. I wouldn't even do it again for myself!