Getting it running again is a real treat, but then you also need to be able to stop.
Here is what I do.
I replace the booster and the master cylinder. I have had pretty good luck going to AutoZone and buying the rebuilds there, and Booster Dewey is well regarded for his work. Just make sure that the rebuilt unit is the same type and style as the one you take out. This is critical with an FE powered Cougar. The good news, Mustang and Cougar are the same.
While I have the booster and master cylinder out, I remove the rubber brake lines. There are three of them: one at each front wheel, and one at the rear. Again, same as the Mustang. This really is not optional. The old lines fail from inside out and they have a funny way of working intermittently: one minute they let fluid flow and the next they are completely stopped up. Just toss them. Your life, well at least the car, is worth more than a set of brake hoses.
With the master cylinder and the hoses off, I use compressed air to blow through the steel brake lines. I put a brake bleeder on the outlet end of each line to catch the fluid and debris. If I see a lot (really just a little is too much) of rust coming out with the fluid, then I will replace the steel lines. This is the easy time to do this. The front lines are the same as Mustang, the rear line is Cougar only. You may find that the proportioning valve will not pass hardly anything. This means it needs to be rebuilt. You can get a kit from West Coast Classic Cougar. They have a good video that I recommend watching before rebuilding the valve. I think it is a good idea to always rebuild these, but I know of many that have gotten by with out doing so. If you have an FE, then you really need to think about doing this as it is VERY hard to get to with the booster installed.
Do not use old brake linings and pads. They will crack up and can bunch up inside the drum or the pad can come right off. New parts are available every where. I recommend that you take the old parts in with you to make sure you get the right width shoes and drums, and so on. You will sometimes find that people have used Granada parts or some other substitute to modify a car. You can’t blame the parts guy if Cougar parts don’t fit Granada brakes.
I remove the front calipers, and the front rotors or drums. Typically, I buy replacements from some place like Rock Auto. The old ones can be turned if they are not too thin, but old iron gets brittle and the replacements are usually in the under $40 range. Seems like cheap insurance, compared to the price of a new bumper. On disc brake cars, I take the calipers apart. (hint: to get the pistons out, I use compressed air, if they are really stuck you can use a grease gun, but if they are that stuck you are probably better off using rebuilt calipers) Caliper rebuild kits are cheap. BE CERTAIN TO PUT THE RIGHT CALIPER IN THE RIGHT SIDE. On drum brake cars I replace the wheel cylinders with rebuilt units. In both cases I buy a hardware kit to replace all of the little parts with nice new stuff. All of the rebuilt calipers and kits and slave cylinders and hardware kits are still available from the chain auto parts stores and places like Rock Auto.
I think this is also a great time to replace or at least clean and repack the front wheel bearings. Nothing sucks quite as much as discovering that the spindle is all munched up and the bearings are bad when you go for that first drive. You have to disassemble everything you just finished to fix the problem.
The rear drums can be tough to remove. Be sure to loosen the brake adjuster star wheel to let the shoes retract away from the drum. I put a lug nut on one wheel lug. This keep the drum from flying off the car when you are pulling on it. If it doesn’t want to budge, I use a small piece of oak, held against the back edge of the drum, that I tap with a BFH. Work your way around the drum. I use a little shop vac to clean things up inside. Replace one side at a time so you have the other side as a guide to help you figure out where every thing goes.
Bleeding the brakes is critical to get all of the air out. I start with the left front. then the right front, and then I verify that the flow is good at the hose location at the rear axle. This means that the proportioning valve is working. Tighten that up and then move to the left rear and the the right rear. I put the car up on jack stands and adjust the rear brakes so that they drag just the slightest amount. I also verify that the parking brake works and in particular that the springs are releasing the brakes.
It should go with out saying, but be certain to remove all of the grease from the rotors and drums. Just spraying the rotors with spray wax can make them slippery enough to render the car unstoppable.