Travelling back to UK on Indian passport with British citizenship

Hi,
I visited India on Indian Passport on vacation for 1 month on 1st April 2018 , I have got ILR but my sons UK visa was expired in January 2018.
Neither at UK airport not at India airport, no immigration officers even noticed that his visa was expired.

Now before going back to UK, I have applied for his citizenship.
I have two questions
1. Can I travel back to UK with his British citizenship certificate and Indian passport with no valid visa
(Does British citizenship certificate means right to enter Uk is valid)
2. Can Childs mother or father apply for his passport from UK to speed up if 1 is not an option

Any help is much appreciated

Many Thanks
Yash

The story is not clear. You and child went to India on expired UK visa of son. As long as your son has valid indian passport, UK authorities and Indian authorities will let you board the flight from UK to India and let him enter India.
Where is son now? India?