How many Bluetooth modules can work in one room?
...without them interfering with each other


Two.

Bluetooth devices change their communication frequency at about 1600 times per second. That is on purpose so that it is less of an issue when there is interference on some channels. Bluetooth is designed with the assumption that data packets will get lost and have to be re-transmitted.

When only two devices are communicating they are "in sync" regarding the channels that they use. Add a third device that is not "in sync" and that will transmit (to show its presence) at random channels. So collisions will happen. That's OK as it will not happen that often.

You ask for no interference and with Bluetooth that's not possible with more than two devices unless these would be "in sync" as well. But then not all devices would be able to communicate continuously, they would have to "take turns".

What you're asking (no interference) isn't how Bluetooth is designed. It is designed such that interference is expected and part of normal operation. That allows many devices to be used in a small space. More active devices will simply mean lower data transfer rates. But the system will continue to work which is an improvement compared to systems that are less immune to interference.