Seem to remember that Margaret Thatcher tried the same with the military when UK ATCOs took strike action in the 80s(?).
To her surprise she was informed by the civil ATCOs that she was welcome to bring in the Mil but the following caveats would apply:
1. Military controllers do not have civil licences.
2. as a consequence the insurance companies would not cover the risk
3. When the first airhit occurs, as it indubitably would, the buck stops with her
and last but not least,
it would require an act of parliament to override the Regulatory Authority and allow said Military operations.
She backed down.
However, despite the risks involved a well known UK Tour company's flight wing flew in uncontrolled airspace under the UIR to the UK/French FIR boundary and then free called for a joining clearance from either Brest or Paris North as applicable.
Good luck
