Nocturnal Mystery Bird AKA "Mr Squeaky"

Hi.  Can anyone make a suggestion for the identity of the bird that we hear at night calling in the fields and woods surrounding our tiny Devon hamlet?  Can't see it (it's dark!) and it seems to call before flying from one perch to another .. we can hear a faint reply sometimes.  Its call is a long drawn out and quite loud squeak; nothing like the "twit" or "twoo" of the owls, although we often hear both barn and tawny owls - I've listened to the nightjar recording on the website, and it's not that either.  In tone it's not disimilar to a blackbird's warning call, but just one shrill note, there's no variance or undulation I can pick up on. My daughter has named it "Mr Squeaky, The Squeaky Bird".  It's driving me mad not knowing, but as a wise person once said, "You don't know what you don't know"!   If anyone has ideas for what it might be, I can listen to the calls on the website and hopefully ID it.  I will report back if I find out!