Turns out I didn't have as much spare roofing felt as I thought I did, so would have needed to buy more, plus the state of the existing roof means laying new felt on top of it probably wouldn't be the best idea anyway.
So, after much deliberation and internet research, I took a different approach - went over the most damaged bits with either roof sealant or Gorilla patch tape, then painted the whole roof with Black Jack weatherproof paint.
The paint is mad stuff - has the consistency of porridge and you need to keep stirring it (really could have done with a paint version of a cement mixer). Also glad I bought a cheap disposable brush for the job - cleaning it out afterwards would have been far more trouble than it's worth. Anyway, I finished the job this afternoon and it has just started raining, so I guess I'll find out soon enough if it has done the trick. Will keep you posted, in case anyone is interested.