Memberships don't get updated when they elect to auto-renew
just to confirm some basics since your question doesn't specify if membership status update is happening for other situations.

For Membership Status to auto update you need to have the Scheduled Job enabled. You can see this on the demo site here. And you need to have a cron running. Neither of these may be the issue you are facing, but are fundamental to the membership status update working.

Otherwise you should confirm your Membership Status is not 'overridden'. Again this is something you have probably checked but it helps if you can confirm such things.

It may help if you can clarify the Contribution Status of the payment. Is it being set to Completed automatically?