I will be moving to the UK in August with Tier 2 long term visa. I'm an Indian passport holder. I would want to get a Schengen visa as soon as possible. How could I get one without waiting for 3 months after getting a residence permit? Is there any way to get the visa as soon I get the RP?
I believe you misunderstood the visa requirements. According to VFS Global:
UK residence permit (original and copy): valid for at least 90 days beyond the return date of the trip to the Schengen area.
Which means you need your residency permit to be valid for at least 90 days after you return from the Schengen area, rather than the opposite way. So you can go ahead and apply without waiting.