We wish to take my mother on a surprise holiday. She has an Indian passport but has indefinite leave to remain in the UK.
Is it possible to go to France for example without a visa?
If not, how long does it take to get one?
Your mother will need a Schengen visa despite holding Indefinite Leave to Remain.
ILR gives the right to come and go in the UK, but it is not recognized as a travel document for Europe. Once a person has ILR there are only a few hurdles left to get naturalised and apply for a British passport.
On the plus side, your mother will be able to use her ILR stamp to apply at any European consulate (that has an visa section) in the UK to get her Schengen visa. There's a high probability that they will issue it also because her circumstances in the UK are sound.
For you question about how long it takes, it depends upon the nature and complexity of your mother's personal circumstances. Normally, I would estimate that 5 - 8 working days is fully sufficient. I need to add... don't buy tickets or make any kind of binding reservation in advance of receiving their decision.