Hi Gill,
I haven't got around to getting to the archives in Solingen yet, but had another intensive look at various census.
I have found the following:
1901 Census RG13, 430, 116, 40
52 Dulwich Road, Lambeth, London
Nathanael G E Muller (head) Single b. 1842 Skin Merchant / German Subject
Franciska Muller (sister) single b. 1863 / German Subject
ANNIE MULLER, NIECE single b. 1886 / German Subject
Mary George, servant b, 1874 domestic servant / Kent
Not sure if this is your Annie, birth date is not quite right but worth thinking about it.
1901 Census RG13, 107, 78, 44
25 Manchester Street, St. Marlebone
Annie Keough, head, single b. 1857 Lodging House Keeper, Ireland
Mary Cahill, servant, single, b. 1884, domestic servant, Norwood
William Muller, servant, single, b. 1882, domestic waiter, German Subject
Again, birth date not quite right, but also worth thinking about.
Viv