If they are rusting away to pieces no company is going to accept them as part exchange?? Look at computer trade-in or camera trade-ins,the trade-ins need to be in working condition. If they are in good condition they will be worth decent money.
In case of the car Zak33 mentioned it was in entirely mint condition. In the end its not really an excuse,to scrap a car worth more than the steel value,and it seems Ford agrees as they won't be scrapping perfectly functional old cars. Cars which are scrapped more recently are unlikely to be the cars you talk about.
The fact a mint condition car from a company which has long gone was being sent to the scrapyard for a Transit,hints very strongly as this being a car inherited from a family member who did look after it and they owner had no clue it was probably worth a few grand.
I know not a single person who would have a 60 year old car in mint condition with everything working who would be scrapping it. The fact its survived 60 years indicate it has been looked after by a car enthusiast and probably an older person.
For example our friend had an MG Metro,which was one of the higher performance models and he got a reasonable sum for it and I suspect most would say its a Metro and get £50 in scrap for it. I mean look at the values of 205 GTI 1.9 models for example. A lot of people simply don't know the value of some of these cars at all if they inherited these things since they CBA advertising it.
Some of those 80s hot hatch Fords are worth a lot now,since so few are left as so many were crashed. They are hardly considered a Bugatti or something.
Its the same in the UK when it comes to some French cars like Venturis and Alpines,which are rare and they are being reverse exported back to France,etc since there are French buyers looking on the UK market for cars.
IIRC,there has even been the case of US cars from here being exported back to the US.
In the end it is still part of our motoring history,and you could argue the same of a Spitfire,etc.
I give you another example during a recent gun amnesty(I think it was here),someone handed in an antique gun(a few 100 years old) which was obvious a family heirloom. Another one was a FP45 Liberator pistol which is very rare. Some of these are worth real money as antiques or at least a donation. Obviously the people who handed them had no real clue of what they were worth.
The police did the right thing and will donate such guns to museums and not melt them down.
So why shouldn't be any different for cars??
If you don't want an old car,advertise it even for free and I am sure people will take it off your hands,even for parts.
Edit!!
I give you a prime example of what I said earlier.
The Vauxhall Chevette is rare - less than 500 left in the country.
I knew someone who had a mint condition one,with a perfect interior and perfect body which had actually been inherited from a friend. It actually won an award in a classic car show years before. It was a perfect runner and had no issue passing an MoT.
The owner decided to go to car dealership who didn't want,it so paid £50 to scrap it.
It was worth £1000s of pounds especially in that condition.
Less than 500 on the road now.