The best way to rejuvenate a really overgrown hawthorn hedge is by "laying" it. This can be done any time of year but was traditionally done on farms in winter, when there was not so much other work to do. It is easier when there are no leaves on the hedge, and these days it is illegal to do any commercial hedge trimming work during the bird nesting season.
The basic tool is a chopper (not a saw). In my part of the UK they were called "billhooks" because there the blade had a curved end, which was handy for pulling branches into position.
Prune off any long branches to leave a hedge about 6 feet tall. At this stage it will probably look a complete mess, with big holes between the separate plants at ground level, but don't worry about that.
Then, chop through about 75% to 90% of the main stem of each plant close to ground level, and bend the top part down almost horizontal. The top part is held in place with stakes, and by "weaving" some of the long pruned branches into it. You want to end up with a hedge about 3 feet tall where almost everything is close to horizontal, not vertical.
A chopper is a better tool than a saw, because it naturally follows the grain the wood to produce a cut that will bend through 90 degrees without breaking.
So long as you leave some bark and sapwood intact when you chop into the branches, everything will re-grow. During the first year, cut out any long growths that are heading in the wrong direction, e.g growing sideways out of the hedge, or getting too tall. By the second year you will have a solid hedge that you can keep in shape with a trimmer.
Warning, this is hard work. If you are a beginner, laying about 5 to 10 yards of hedge in a full day's work will be good progress.