I am looking for some ideas for a low cost circuit to detect if an antenna
has been broken or removed
from a 10mW transmitter.

Hi David, The best, low cost idea that comes to mind is look and see if the
antenna is gone.
I don't think that was what you were looking for. What you need is something
like the circuits in solid state HF rigs that reduce power when they sense a
mismatch. At 10mW you will need additional circuitry to boost the signal up to
a level that the sense circuit can use.
All the circuits I know about will cost quite a bit more than your 10mW
transmitter. Out of curiosity, what are you trying to do?
73 Gary N4AST