Wow, there are some snarky people on this thread!
Businesses don't exist to provide convenient jobs for people. The OP's primary responsibility as a manager is to ensure that the business is sustainable. At the moment, it isn't because the flexible working arrangements are not meeting the needs of the business.
If people aren't willing to change their hours voluntarily, and the current working patterns cannot be sustained, then clearly, the manager will need to take action. The most obvious way of doing this would be to go down the "dismiss and re-engage" route. That is, you terminate existing contracts and then re-engage on new terms and conditions. You would need to have a proper consultation process before making these changes, and of course, some employees might choose to walk away.
Obviously, before taking such action, you'd need to be pretty sure that the business needs were real, as there is no point in pissing off your workforce without a very good reason. I'm all in favour of flexible working where it's possible, because happy employees generally make good employees in my experience. However, there needs to be a balance and it needs to work operationally.
FWIW, I say all this as someone who has worked flexibly for years and who is currently facing redundancy, so I'm not only speaking from a manager's perspective. As the OP has said, if she doesn't make some tough decisions, the whole business might go under in any case.