Schengen Visa | VFS - appointment scarcity

Looking for some advice, I am travelling from India to UK in July to visit my brother. I will stay with my family for a month in the UK. Standard visitor visa is already granted, no issues there.

After that we will travel to Europe for 25 days to cover five countries - France, Netherlands, Switzerland, Italy and Spain. We are staying in France for more days as compared to other countries, so I understand Schengen visa would have to be filed from French embassy via VFS in India

I currently have an appointment with VFS, Delhi for Monday, 13th June to file the Schengen visa application, however my flight for the UK is on Tuesday, 28th June. There is only about 10 working days between 13th and 28th June. I am not sure if this is enough time to get the visa. The issue is with VFS as they don't have an earlier slot available to file the visa as I was looking since May and as far as I know they don't entertain walk-in.

I have spoken to a few travel agents in Delhi, most are saying Schengen visa usually comes within 5 working days, but I guess that would depend on a number of factors as to what time of the year etc etc.

Just want to check if anyone recently has applied for Schengen visa from France Embassy in Delhi and how many days did it take to get the visa?

Any help, advise, suggestion is really appreciated.

Many thanks