Like the Kensal Rise Library, Cricklewood Library has succeeded in fund raising for its capital costs. The next part is securing the running costs of whatever is in there. In this, I would have misgivings about whether trying to run something resembling a local authority library is sensible. The infrastructure of running even a small local authority library is extremely expensive and cumbersome.
What I think many of these groups actually want is a community space and a symbol of the local area, which could be obtained far more cheaply since it would not require things like computers, staff/volunteer training etc. that add considerable complication.