On FreeBMD there is a marriage between John James Coote and Ann Evans, Sep quarter 1880, St Olave 1d 360.
Also, I saw a Mary Ann Coote birth in Bethnal Gn, June quarter 1856, 1c 254. Maybe that is the registration name to look at.
From the marriage register St Mary Magdalene, Bermondsey
August the something, there is no date, 1880 (the previous entry is August 2nd)
John James Coote, 40, widower, Butcher, of ? Place, Snows Fields, father William Coote, Carpenter
Ann Evans, 28, spinster, of 147 Snows Fields, father William Evans, Farmer
John James signed his name with a flourish, Ann made her mark
witnesses: John R? and Elizabeth Evans
Some of the Westminster baptisms have been transcribed and put online=, I had a look for your family but there are 2 other Coote families in the area at the same time, James & Eleanor (he's a sweep) and Charles & Charlotte (he's a plumber).
Remember that although civil registration started in 1837 it wasn't compulsory for births to be registered so there may not be births to find. The ones you identified earlier may belong to the other families.
26 April 1857 St Mary Magdalene Woolwich
James Coote, full age, bachelor, Porter of Callis Alley, father William Coote, labourer
(fathers occupation and James signature don't match 1880 entry)
27 Feb 1859, Bermondsey, Father = Mark, a dyer, signature doesn't match
28 July 1861, Chelsea, James is a baker as is his father James
I couldn't easily see the others, they may not be Anglican marriages.
Anyway hope this helps.