How can I set up multiple payment gateways on my e-commerce site so payments route based on the customer’s location to the nearest branch?

Look at super-aggregators like Adyen, Wirecard and I would highly recommend BlueSnap (payment processor) (no affiliation to anyone of them).

Companies like the above have payment processing facilities (integration) with like all the payment systems worldwide that matter, so if your customer is from Singapore and wants to use a payment system native to Singapore, these companies can provide it - for all the countries in the world, in a single API/Plug-in.