This has got me looking into some of my books on the subject, principally because I was sure that I had read of Hudgillburn (all one word) Mine and I've found a mix of references some with one word, and some with two!
The reopening of the main level by CATMHS was written up (by Sheila Barker) in "The Mine Explorer" (CATMHS's journal) Volume 5 dated 2002. In that volume, Sheila extensively uses the one word version of the name - Hudgillburn - including not just the mine itself, but also the name of the level that accessed it, and that of the company that worked it). However, she also refers to R.A. Fairbairn's monograph "The Mines of Alston Moor" (published as British Mining No 47, in 1993). I don't have this, but in the 2nd edition of the same (published in 2008, and seemingly outside of the British Mining series) it is referred to as two words - Hudgill Burn Mine.
So possibly either form is correct?