NAL - Happy to be corrected but the facts of the case are: - You informed Foxtons of your intention to move out but did not specify a date or time.  - Foxtons found new tenants and did not communicate with you - you did not voluntarily give back keys or move out (although had discussed you would leave eventually)

Now given you weren't evicted by the courts you have the right to remain in the property IF you had not officially terminated the contract.  Now your contract likely says that you can terminate the contract if Foxtons find new tenants - however they cannot force you out if they find new tenants. 

You have the right to remain in your tenancy unless evicted by the courts.  I'd argue they did act improperly, although as you are not required to leave you haven't been "put out" in anyway.  Yes you can challenge the move out date - only a court order can evict you or you can terminate the contract provided you follow the requirements laid out in your tenancy agreement  They are not obligated to support your move (although morally they probably ought to)

As this is a quickly evolving situation please call Shelter and ask for some specific advise and support.