EDIT January 10, 2021: This answer deals with Q1 about Parental leave, and compensation for the leave-taker. Traveller's comment below addresses Q2 regarding fiscal support by the UK for a birth abroad. It notes that such support is not available if the birth occurs in India.
The UK Guidance on Paternity Leave, Eligibility, and Pay do not address where the leave-taker must be (or not be) during the leave period. This means your physical location during the leave period doesn't matter.
However, note that the "Leave" Guidance includes this:
Start and end dates
Leave cannot start before the birth. It must end within 56 days of the birth (or due date if the baby is early).
You must give your employer 28 days’ notice if you want to change your start date.
You do not have to give a precise date when you want to take leave (for example 1 February). Instead you can give a general time, such as the day of the birth or 1 week after the birth.
Leave may begin only at or after birth. Thus, you would have to select a leave commencement at birth, or x days afterwards, and unless you'd been able to schedule regular vacation before the birth, would be unable to travel to India before the child is born.
The "Eligibility" Guidance says this:
be employed by your employer up to the date of birth
earn at least £120 a week (before tax)
give the correct notice
have been continuously employed by your employer for at least 26 weeks up to any day in the ‘qualifying week’
The ‘qualifying week’ is the 15th week before the baby is due. This is different if you adopt.
If you usually earn an average of £120 or more a week, and you only earned less in some weeks because you were paid but not working (‘on furlough’) under the Coronavirus Job Retention Scheme, you may still be eligible.
Finally, the "Pay" Guidance says:
The statutory weekly rate of Paternity Pay is £151.20, or 90% of your average weekly earnings (whichever is lower).
Any money you get is paid in the same way as your wages, for example monthly or weekly. Tax and National Insurance will be deducted.
Start and end dates
The money is usually paid while you’re on leave. Your employer must confirm the start and end dates for your Paternity Pay when you claim it.
To change the start date you must give your employer 28 days’ notice.
This suggests that payment is handled by your employer. Whether your employer can or will process payment(s) to you while in India is a matter for your employer; payment to a leave-taker overseas is not prohibited by the Guidance.
Note, however, that for a maximum leave of two weeks trying to arrange such overseas payment may be more trouble than it's worth.