Offer the first piece of felt up to the eaves, with the back of the felt facing outwards, and draping down. Fix it with the felt nails, then fold the felt back over on itself, up the roof. Rinse and repeat.
You can experiment with a piece of paper to understand what I mean.
Sheds are often covered with 1 layer of cheap thin mineral felt, which doesn't have the thick bitumen layer on the underside to torch (it's usually type 1E felt, and was designed to be laid using hot tar). Yours looks like this to me (as is my own summer house). However, since it's intact you could use it as a first layer by adding more clout nails to it, then torch a layer of torch-on felt over it. This would give a roof which should last 20 years or more.
Torch-on felt is £50-70 per 8 metre roll at builders merchants but can be found from about £25-35 on fb marketplace etc. It's difficult to tell but I think you have 4 strips, each probably under 8 metres long, so you'd probably need 4 rolls, plus a short shift from a roofer someone knows from the pub.
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