I've had packages take over 3 weeks to get to the UK or Canada, and I can only imagine it would take way longer than that to get to India. USPS has vague quotes because once it leaves the US it's beyond their control, just like once it leaves your hands it's beyond your control.
I've started getting insurance on all international packages that I don't want to have to either refund or remake. Unfortunately insurance doesn't reimburse for returned packages, so I no longer ship to certain countries that have been consistently challenging with packages being lost or returned to sender (the UK in particular, unfortunately).
In this case I would tell my customer that I'm attempting to file a 'lost or delayed mail claim', which you may not be able to do. They should also file such a claim with their mail service. I wouldn't plan to reship or refund until at least a full month or even two months post-shipment. It really can take that long...
You may want to update your shop policies for the future.