Planning to get my mother on 02 year family visa category to spend some time with my sons who are both born in the UK and british citizens... would like to know how long will it take for the visa to process...
Processing times for visas vary according to the location where the application is made, the strength of the application, the number of other applications made at that location at that time, etc.
You can get an estimate for your particular case from the UK Government web site, which is currently giving estimates based on actual processing times for applications made in December 2018.
A quick look at a few random locations suggests that most visitor applications are decided in 15 to 30 days. Your mother's application could be quicker or slower than this.