The namaz timing of Zuhr as per many calculation we receive online says it is 12:15 in bangalore India. Therefore can we pray right after that zuhr & Asr or wait for azan which happens in the mosque at 1pm. Mosque is far away I won't be able to pray jamat. So what's the condition can I finish prayers at prime time 12:15 or wait for masjid azan and then read?

Yes you can perform your prayer straight after it’s actual timing, as you don’t need to wait till it’s performed in the Mosque