The roof has done quite well to last 20 years, I would remove it and lay a new roof that will have a ventilation barrier and vents fitted to allow air flow.

It sounds like there was no vent sheet i.e. first layer should not be stuck so bitumen should not be in contact with boarding.
You may be able to stop most of this by painting the roof with a solar reflective paint. Aside from that, being that it's 20 years old, maybe you should look at replacing it.

You should consider moving away from a cold roof construction and look towards having a warm roof fitted with no ventilation as per current regulations.

屋顶有一个预期寿命等一切and the life of this one seems to be at the end . My advice would be to get some reputable roofer /builder to clean all roof off then give prepared area 2coats of solar reflective paint which would help to stop the bitumen from melting and dripping through board joins this would give the roof maby up to another 5 years of life . But in the long run I would advise a new roof covering and there are many different ones out there from the old style cheaper green mineral hot bitumen covering to a new g r p glass fibre one that have up to 30 years guarantee. The decision is yours