There are three methods I usually try, in order:
1. Drill out the edges of the holes in the plastic slightly, so that you can push the removal keys in at a steeper angle. Sometimes this is all it takes to let you engage the clips and pop it out like normal. This usually doesn't work, but it's worth a try.
2. Try to reach up under the dash or through the glove box, and press in the clips from the back. This usually results in scrapes and bruising on your hand and forearm, with no success in removing the radio, but I usually try it anyway, because the next step is
3. Tear apart the front of the Ford radio until you can get to the clips with a cotter pin hook or a pick.
This is why I'll never try to remove a Ford radio with anything but the proper removal tools. Once those clips get bent out of place, the easiest radio removal in the world turns into one of the hardest.
Good luck!