Banks (and others) won't accept a death certificate from anyone other than a relative as proof of a right to access the funds they hold. With a very small estate they
might accept a death certificate from a relative but they'll usually require a grant of administration issued by a Probate Registry. (That's either probate or, where there's no will, letters of administration).
Further, since the couple weren't married, the gentleman has no automatic right to his partner's assets. In the absence of a will, the laws of intestacy apply, as illustrated in this flow chart (where 'you' refers to the deceased lady):
http://www.tollers.co.uk/system/docs/178/original/Rules%20of%20Intestacy%20Flowchart.pdf
The man also has no right to apply for a grant of probate.
As things stand, the man has no right to receive anything from his partner's estate. Only relatives (as shown in the flow chart) or, in the absence of such relatives, the Crown can benefit. However if your neighbour believes that the intestacy rules will deprive him of 'reasonable financial provision' from his partner's estate, Section 1(a) of the Inheritance (Provision for Family and Dependants) Act 1975 allows him to seek a court order varying those provisions:
http://www.legislation.gov.uk/ukpga/1975/63
So it looks very much as if he'll need a solicitor.
Chris