Hi Dom. A quickie search at
http://freebmd.rootsweb.com/cgi/search.plfor surname Melhuish born 1875-1890 shows only two with first given name May:
May Pearl Melhuish, Haslingden, June quarter 1879
May Melhuish, Chelsea, June quarter 1884
Of course the questions are: was May a nickname? was Melhuish her birth surname or a married surname?
Neither of those seems to be in the 1911 census, so presumably both had married (or died).
May Pearl married in 1900, for one.
I wonder whether the Chelsea May might be the one who married Herbert W E White in Hampstead in 1918. Those are potentially 'privileged' locations.
A May A Melhuish aged 28 (born c1883) died in 1911 in Wandsworth. I wonder whether your portrait might have arisen out of a young woman's death. She would most likely have been the Mary Ada Wisdom who married the marvellously named Canute Ridge Melhuish in Wandsworth in 1900. In 1901 in Gravesend she is called May Ada Melhuish, and she was born May Ada Wisdom in 1879 in Kensington. He was a cycle maker, but he was later in a theatrical occupation, and May was not in the 1911 household - perhaps being ill.
The two couples, White and Melhuish, both had children. The Melhuish-Wisdom children can be seen in the 1901 and 1911 censuses, and the White-Melhuish children by searching at FreeBMD for births with those surnames.
Those look like the most likely possibles!