What should I do when the references are not replying?
People get loads of emails and it is very natural that some take a few days to reply. Also some emails may be missed or overlooked, so you can write again and say it's urgent. Normally such things should be done with a few weeks time if at all possible because answering to such requests is not the first thing on people's list.
If all else fails, try to contact somebody else. I'd avoid to give Professor X the list of names without having an answer from them. It doesn't look good to pass on such information without being authorised. You can ask Professor X for advice what to do without telling them names. Surely if Professor X is responsible for the tight time frame, they should have some understanding.