When I was enquiring in the 80's the Mother house at Mount Vernon had closed down and the Sisters were at Broughton Hall. However I believe they moved to a smaller house in Finch Lea Drive, Liverpool, in about 1993 . There is an archivist in London email: firstname.lastname@example.org
Tel: 0207 232 4882
Institute of Our Lady of Mercy Archives
Convent of Mercy
London SE1 2DQ
It's worth a try - it can't do any harm.
The only other place you could try would be Catholic Social Services
Nugent Care, 99 Edge Lane Liverpool L72PE
but I think they dealt mostly with adoptions.