@StarBear are these the pieces of vinyl you are talking about?
Click to view attachment photo of mine when i was going through it recommissioning a couple of years ago.
i recall its very thin vinyl. unusually thin.
probably so it could take up all the different contours.
it was all curled like yours at the edges.
i washed them in warm soapy water. got them all clean.
because i did not want to use any "pollutants" that might affect a later glueing process.
i glued them all back down again with a contact adhesive.
using the double tack method. ie i also sprayed glue in the area of floor i was glueing to.
(i marked it out and made a paper stencil to stop overspray.
it appeared to hold the edges down which had curled tightly as you describe.
mind you i have not had a look to see whats happened since.
its probably curled again. not sure.
the problem with the vinyl is that like all plastics it gradually ages and loses its plasticizers. i think that is the generic name for the compounds used to make flexible plastics.
i'd be cautious using heat as it often has a result which exaggerates the curling.
ie it makes it de-plastize even further.
another solution would be to find some vinyl sheet from an uphostery source that matches the original and use that.
however i think it will be hard to find sheet vinyl as thin as the original stuff.