Hmm!
Usually a broken stopcock lid in the road, is the water authorities problem to free. You could report it to them.
But have you tried, if its a cast iron grid top, tapping the hinges with a hammer? If you angle back the hammer and knock back wards and down wards just past the pivot point of the hinge, it often free's the lid enough to get the tang of the water key under the front edge.
You can of course freeze the pipe inside and change the internal stopcock. But in some houses there just isn't the length of pipe before the stopcock for you to be able to do this. In which case its back to the outside grid top job.
You could at a push and if the external stopcock isn't on a path or where somebody could fall down it, lift the flagstone and get at it that way?
But don't forget its your job to reinstate the ground around the stopcock and look after it for a few months afterward.
I would get the local water company to do it if I was you. Its usually free by the way. But if the house is flooding out and the job can't wait.