Churches don't have dungeons, which are associated with castles -- the word is a variant of donjon, the great tower or keep of a castle which had underground storage areas. The word later came to be applied to cells and prison-like areas under the keep. Warwick Castle has particularly fine dungeons, I believe.
Larger churches often do have underground areas, but these are known as crypts, which were often used as chapels or places for burial. Many English cathedrals have crypt areas, some of which are open to visitors. One which I know is Winchester, which can only be visited as part of a special tour -- the water table is very low there and it's often flooded in winter. Here's a copy of an image of the most often viewed part of it, which can be seen from some steps which are accessible from the cathedral -- the vaulting is very typical and is repeated throughout:
What isn't immediately clear is that the crypt actually extends a long way to the right of this. I've only been down there once, but from memory I think there were some air/light shafts linking to the outside, coming out at ground level. It's possible that in the past they could have been used for someone to escape into or out of, but it would have been something of a scrabble, but nowadays they're blocked, so if you're setting your story now or in the near future, this would be a problem.
A crypt might also be known as an undercroft, as at Durham Cathedral. Here's how it looked in 1893
Crypt Durham Cathedral but again it's difficult for you to use it as a place to escape today, because it's now a restaurant/cafe.
It's often the case that the crypts/undercrofts have been repurposed as cafes, shops or exhibition areas because of the need to attract tourists and increase revenue. It's possible that somewhere in England there's a large crypt which is unused, but it's less likely to be mysterious than simply very damp and cobwebby. I've no idea what the position would be abroad, though.