Hi,
Ohhh this is a very interesting family. You are now dealing with the "top brass" in UK

OK - you have a winner!!!
Caroline May Gladstone was the daughter of John Hall Gladstone. He had at least 4 daughters. Another one was Margaret Ethel Gladstone who married James Ramsay Macdonald (the PM!!)
Caroline May was born in Kensington Dec qtr 1863 (ref 1a - 90)
Margaret Ethel was almost 10 years younger than Caroline.
John Hall Gladstone was a Scientific Chemist and a lot of info has been written about him.
He was the son of John Gladstone and Alison Hall. He was born 7 March 1827 in London. He married twice. First marriage to Jane May TILT (M: 1852 in Bath)
Second Marriage to Margaret King (M: July 1869 Edinburgh)
As Caroline was born in 1863, it is assumed that Jane May Tilt was her mother (particularly with a second name of MAY)
I haven't followed her any further than the 1881 census when they were all living in Pembridge Square Kensington (with Butlers/servants etc)
I hope all this has been a help.
If anyone else can add to this I'm sure Forsyth12 would be grateful - particularly seeing this is a "first post"
Happy hunting and good luck with your research
Di