Would a neurologist make referal for TCD Bubble test for testing for PFO in the UK? I do have Migraine Aura 4 a mth, but is the test specifically for Migraine aura WITH the Headache only?

Probably not. The bubble test for Patent foramen ovale is a simple test that is benign. It only tests if there is a PFO, which does appear to be more common in people with Migraine and aura. However, studies have shown no improvement in Migraine after surgery to close the PFO, so most neurologists would not recommend getting the test. See, for example this recent:
https://www.ncbi.nlm.nih.gov/pubmed/26952049.