You can check directly with the British diplomatic staff in India if they are willing and able to carry out the citizenship ceremony there. And tell us of what they advise.
But, as I understand it, the only people who can take their oaths of allegiance at a British diplomatic post abroad are those adults who had an
entitlement to register as British citizens, normally because they would have been born as British citizens abroad, but for historic legislative unfairness (because British mothers could not pass on British citizenship automatically at birth, etc). Essentially their citizenship ceremony abroad merely restores them to a status that they would have been born with.
Naturalisation is an application at discretion, not an entitlement. So it is unlikely that your spouse will be able to take her oath overseas.
Spouses of British citizens on Crown Service are also allowed to take their oaths overseas, because they are overseas at the direction of the Crown/government.
Also, the citizenship ceremony is meant to highlight your integration into the local community in the UK. That is why it is tied to the local council you live at or have some connection with, not any random council that has a free slot.
Also been aware that applying for a first British passport from abroad may cause the passport application to be scrutinized in more detail and so may cause delays.
And while the child born abroad will be a British citizen by descent if either one parent is a British citizen otherwise than by descent at the time of birth of the child, be aware that any children born to that child (your grandchildren) born
outside the UK will not be British citizens automatically. Any children born to them in the UK will be British citizens.
British citizenship is limited to one generation outside the UK.
EDIT: Got the quote from Gov.UK I was looking for. I had read it years ago, but couldn't find it.
Citizenship ceremonies - Gov.UK wrote:
If you’re not living in the UK
You can ask the embassy or consulate in the country you’re living in if you can have the ceremony there instead.
If you’re only abroad for a short time, you might be asked to postpone the ceremony until you return to the UK. You must still book your ceremony within 3 months of getting an invitation.
You might have to prove you’re planning to live in the UK permanently if you’re going to be abroad for more than a few months.
If you applied for citizenship in the UK, you cannot have your ceremony abroad.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.