There's also the term 'swallow-hole' which is a variant of 'sink-hole'. A sink hole is, "...a cavity in limestone etc. into which a stream etc. disappears..." (OERD). Dames says something similar but the problem is that there's nothing going into the Swallowhead - the spring when it's flowing is coming out; that seems to be quite an important difference.
Swallow (v) in Old English and up to about 1000 was written swelgan, then up to about 1380 it was written swolowen.
Swallow (n) in Old English and before 800 was written swealwe, then around 1300 swalu.
In other words, if there's a 14th century map or reference somewhere that shows the Swallowhead Spring written either as Swolowen Head or as Swalu Head, it might settle the matter one way or t'other (I'll go and get me anorak now :-)