Reading between the lines, it seems the old management might not have been great at invoicing people, hence the new mananger being tasked with bringing all the account (not just a mistake with yours then) up to date.
No idea what the law is but I think it's pretty bad form tell someone they owe a couple of grand and three weeks later get the hump because they haven't quite paid it all back. Particularly suspending your children.
If you want to move nursery then do so but if your boys are settled and enjoy it, stand your ground and explain that you've already made great inroads into a debt you knew nothing about. If this firm is a 'bigger' company, is there a head office you can contact?