Ewan, regarding not being able to search for Alice Emily without a surname, you might find it quicker to search the births alphabetically on www.isle-of-wight-fhs.co.uk/bmd/startbmd.htm rather than on Freebmd, and you could pick up the only Alice forename ones at the same time.
I tried A* which brought up all the A surnames for Alice over the 10 years 1867-1856, which then have to be skimmed through (or "find on this page") for ones with a second forename Emily, and registered in Ryde (no Alice Emily but two Alice ASH, Ryde). Your Alice Emily Attrill comes up as ATTRILL, Alice Anderson there, although there is an ATTRILL, Alice Emily in Cowes, same year.
Under "B*" The Alice Emily Bull whom you have checked was the only one, plus BUCKETT, Alice 1864 Ryde. (I ignored the ones with second names other than Emily.)
Under C* in Ryde: COOPER, Alice Emily 1861; COOPER, Alice 1858; COOK, Alice 1858; COLEMAN, Alice 1863; COLLINS, Alice 1865; CORNEY, Alice 1865; COOK, Alice 1866.
Under D* in Ryde: DALLIMORE, Alice 1861; DAISH, Alice 1862; DAMP, Alice 1864; DAVEY, Alice 1864; DRURY, Alice 1866; DYER, Alice 1867. No Alice Emily found.
Under E*: EDMUNDS, Alice 1864; ELEY, Alice 1866.
F*: FEBEN, Alice 1862
G*: GRANT, Alice Emily 1864; GAWN, Alice 1861.
H*: HUMBY, Alice Emily 1864; HAYDEN, Alice Emily 1865; HEWARD, Alice 1858; HATCHER, Alice 1861; HAYDEN, Alice 1861; HAPGOOD, Alice 1863; HAYLES, Alice 1863; HOSKING, Alice 1864.
I*: none
J*: JEFFERY, Alice 1859; JAMES, Alice 1866.
All in Ryde.

So judging by these first 10 letters (which took about 15 mins to check) there weren't many Alice Emily births in Ryde. But of course it takes very much longer to check for their marriages and then in the census.

Sue