Game doesn't recognise xbox controller after changing batteries

I have a weird problem with Windows 10 (build 1903) where if I’m playing a game with my xbox 360 wireless controller, if the battery runs out and I replace it, after that the game doesn’t register any input from the controller.

Note that the controller is recognised by Windows – I get the Windows ‘device attached’ sounds and if I press the xbox button the controller, the xbox overlay comes up. But the game remains oblivious to the re-attached controller.

I’ve noticed this with multiple games so I don’t think it’s a bug in the game. Since people are bound to ask, the first game I noticed this with was Forza Horizon 3 (from the Microsoft store), but Steam games have the same problem.