Yes, apparently they do sometimes nest in compost heaps. I found the following on the BHPS site.
… ‘From BHPS. Gardening with Hedgehogs.
Compost – another ideal place for a hedgehog to make a nest and rear its
young. Take care when turning the heap; one thrust of a fork can easily
kill more than one baby hedgehog. The safest time to spread the heap is
probably Oct/Nov when most babies have left their mum and adults have
not yet started to hibernate. Partly used bags of compost may also have
nesting hedgehogs in them.’ ….
I don’t know about the wood ash. I wouldn’t want to risk it, myself. I would also be slightly worried that some of the males may have already begun to hibernate by Oct/Nov. So I would be careful at any time of the year – there may be other creatures in there as well. Is it possible to do a more careful exploration first? A bit of careful excavating?