WooCommerce uses the Stripe Customer ID and Card token to process payments. When migrating a Stripe account, the Customer ID and Card token will stay the same if you follow the process outlined here: https://support.stripe.com/questions/copy-existing-account-data-to-a-new-stripe-account
Once you’ve done this, you should be able to simply update your API keys in WooCommerce > Payments > Stripe and automatic recurring payments will continue to work. We would recommend, though, doing a test order and subscription to make sure that everything is working correctly.