Start with the easiest thing to check, the flexible lines.
Just a visual inspection should tell you if they've been in there for more then 20 years.
Maybe the whole life of the car.
Sometimes just the front ones will be replaced, but the axle one gets neglected.

If the lines are OK, then rust inside calipers, but if they were sticking, they would stay on and would not let go after an hour.

Then lastly the servo, you can rebuild it yourself, or there is a new unit that is available, but it just doesn't bolt in, plus they have had problem with it, it has a tendency to hold the brakes on just like what's happening to you now, even on new units.

 

There will be lots of suggestions on here as to what might be the cause but at least you have established that it is brakes and all four which narrows down the field.
As Jeff says start with the easiest things which are often the cheapest as well first.
Brakes are an easy thing to understand. You have a pedal which pushes a rod in the master cylinder. The master cylinder compresses the brake fluid with some assistance from the servo. The fluid is held in copper pipes with three flexible pipes on the S Type and I think four on the Mk2. The fluid then compresses the fluid in the brake cylinders which forces the brake calliper pistons out onto the disks or drums. I think you said you had an anti creep device fitted which is in line with the rear brakes.
So starting with the simple check that your brake pedal is not catching or locking in the down position. Lubricate all joints and remove obstructions.

Check the rod and piston in the master cylinder are returning and not locking in the forward position. There would be slack between the end of the rod and the brake pedal if the piston was getting stuck in the down position. Rebuild or renew the master cylinder.

Forget the servo for now as they are expensive but not too hard to change just difficult to get to on an S Type. Can't speak for a Mk2.

Brake pipes are made of copper and if you had a leak your symptoms would be no brakes as opposed to locking on but the fluid is different and we will come to that.

Flexible pipes are the same as the copper ones. Leaks would give no brakes and any expansion in the rubber due to old age would cause weak or no brakes so ignore them unless you are going to change the fluid which we will come to.

The Anti creep device I would either remove or replace. This should only be working on the rear brakes as it is on the copper brake pipe running to the rear wheels. It works via a solenoid which acts like the brake pedal pushing on to a master cylinder compressing the brake fluid going to the rear brakes. There is a switch attached to the brake to switch it on and another switch on the accelerator to switch it off. It should not affect the front brakes so does not match your symptoms but best to eliminate it by renewing or removing.

Brake calliper pistons are forced on to the discs or drums by the brake fluid. If there is any corrosion on the pistons or the bores you might get the pistons locking up in the out position. This is caused mainly by corrosion on the outer rim of the piston not allowing the piston to retract. With your symptoms I would forget about this as to have all four brake calipers go at the same time is unheard of. HAving said that it would be worth checking that all four are not locking out at the same time. If so clean or renew the pistons.

Brake fluid can get water logged over time and this causes spongy brakes and no matter how many times you bleed them the spongy feeling remains. Not your symptoms but brake fluid can also get dirty. Any dirt in the system could stop some of the components in the master cylinder, servo or callipers from returning to rest which might cause the brakes to lock on. As it is all four it would not be an individual calliper but could be in the master cylinder or the servo. I would drain the whole system of fluid and give it a flush to clean it. If you have drained the fluid now would be a good time to renew the flexi brake pipes. Not expensive, easy to do and it would cost you more money on brake fluid if you decided to do them later.Whilst the system is empty I would strip the master cylinder out clean it and replace the seals. Leave the servo for now unless you are flush with money and buy the higher ratio servo that SNG sell. Less pressure needed on the brake pedal.

Working through it logically in my opinion and I am often wrong there are only four things that it could be to make all four brakes lock on at the same time. Brake pedal getting stuck down, brake master cylinder sticking on, dirt in the brake fluid or a sticking brake servo but as I said I am often wrong. As the brakes unlock after time this suggests to me it is a fluid problem not being able to return instantly to release the brakes but able to release slowly in time like a slow puncture.

 

I appreciate everyone's ideas and suggestions. Thank you. I am pretty good with brakes, but not so much experience with the vacuum servos. I have examined the barking system and I don't think it is the flexible hoses or the MC or calipers. Issues with those items are relatively straight forward and not related to heat. So, I am going to focus on 3 things:

1) Brake fluid is hygroscopic. If there is excess moisture, perhaps it is vaporizing with the heat and creating excess pressure locking the brakes on til it cools. Kinda like a brake system vapor lock

2) The brake servo or check valve is faulty and the excess heat create excess pressure on the system.

3) There is actually vapor lock and it is creating an over vacuum situation with the b=rake servo/booster causing the brake locking.

Any thoughts?

 

Agree with Rob. You must go through the system methodically. As you have established that it is all 4 wheels it sounds typical of a brake booster/servo issue.

 

Sorry. We posted over one another. Yes brake fluid is hygroscopic & high H2O levels will reduce the boiling point of the fluid until it boils at the calipers & you loose braking efficiency or total brakes.

I would do a full rebuild of the brake servo. Don't mess around with brakes on a car that has single circuit brakes. At least with dual circuit you don't lose total braking & don't rely on Jaguar handbrakes in an emergency.

Barratts offers a rebuild service at a price which is what I did & it came back looking as new. They also offer cheaper replacements at lower cost but require some minor re-jigging of the pipework. Otherwise you can do your own rebuild. Well described in the service manual. Rebuild kits come and go. Sometimes available. Sometimes not.

I find rebuilding brake boosters/servos a PIA if you don't have all the gear. e.g. if you have to re-sleeve the cylinder.

 

Example of a Barratts rebuild. And yes they come back looking exactly like that. Great job. They require an exchange unit that is rebuildable.

 

It looks like you are in the US, so you might want to contact "White Post Restorations" for a quote as well. Servo rebuilding is listed as one of the services performed. Back when I was "in the business" our shop used them quite a bit on caliper rebuilds for classic Jaguars and their work is outstanding:

https://whitepost.com/wp-content/upl...ctive-2019.pdf

 

Thanks. I forgot about them...used them in the early 80's to rebuild wheel cylinders and master from a 50's Alfa!