Andy there's no cheap way really to do this properly. The stuff that's falling apart is probably what they used to call sarking and likely well past it's sell by date.
The only way to do a proper job is take off the tiles and battens, replace the old stuff with a modern breathable roofing membrane then relay battens and tiles.
Because the membrane goes on top of the rafters and beneath the battens there's no easy way round it.