Looking for early release from project during notice period

Hello Friends, It's going to be a month since I put papers down and the same day, I had a word with the manager and onsite guy for releasing me early citing the challenges in my life and they assured me of having a word with management about my early release. When I got no response after days, I followed up further and they again reassured me about my release. I even suggested going with a buyout and adding me in the fluidity list as they've made 2-3 resigned forcefully( those people were a part of that list). Meanwhile I had sent an email to HR and she asked for project approvals which I don't have. Today, I found out that as per TCS standard they can't give release as per my manager and asked to talk with the Ultimatix manager. Now I'm back to step 1 after a month. My joining date in another company is just a week away. Another company's HR might extend it for 10 days on the condition of joining them only with no counter offer, plus I've still two months to serve, I doubt they will wait till then.

I want to be released before mid December. Kindly share how it is possible to get early release, I'm ready to do whatever is required.