Would I get a refund for a false diagnosis on my car?

I had a squeal coming from my car and I was never confident it was correct. I got charged £70 for them to make the diagnosis it was my exhaust manifold gasket and I declined them repairing it and took it to another garage. The garage asked if I had my auxiliary belt changed recently, I said yes, and try said the steering wasn't put back properly when I had it done. This then damaged my car. Ford said they'd pay for it to be repaired as they messed up.

As for the diagnosis, should I also ask for the £70 back from that. It was a week before I got it looked at elsewhere.

Go and complain to them. You might be pleasantly surprised. But if they are the clowns I fear they are, even standing there and yelling at their customers won't help.

You can at least let your friends and enemies know why they are highly NOT recommended.

I see there in your question, a story about a "squeal sound" diagnosed as an exhaust manifold gasket.
I have a problem trying to make the connection there.
It may be a separate thing.- a glaring issue?

Then I read "auxillary belt?" Then mention of the steering not having been put back properly, then you drove like that for a week.

Then there's mention that Ford caused you a damage, and said they messed up. They agreed to correct it.

Now, your question must be… Do I need an exhaust job? Or not?
What does that have to do with a belt squeal?

While Ford is correcting their work, and getting the car right,
Shouldn't that issue present itself as well?

You can trust and communicate to get the jumbled information, 2 shops and more than one visit figured out?
Or, you can correct the separate issues, separately.
Does re-evaluating the Squeal sound prove it is solved?

Then, does there still seem to be a different, exhaust issue? Unrelated to the squeal?
You say the exhaust diag was 70 by itself, but you did not authorize
That?
You were there about a belt and the subject was changed to exhaust gasket by your description.

Take your repair orders from every visit to both and any third shops to the Ford place
and negotiate and communicate the confusion.
Here in the states, they have a reputation for expertise in separating and dissolving your concerns. With corrections.
I hope they treat you just as well wherever you are.
That company has satisfaction telephone numbers to call for direction
about unresolved or confused people and repairs.
In fact, here, they call you to see how they are doing, and pay people to find out your mental state.
Part of your issue(s) can even be covered by a warranty? Campaign?
emission recall? We can't answer fully without any information.
I would let them have the evidence and the paperwork to solve it.
To them, in the states, a response of "dissatisfied" requires their full attention.

The apples go in one basket, and the oranges in the other.
The blame and confusion or calling one the other should be left at home in the trash.
If you get sent to where the car gets corrected, I expect the Ford servicebay to be the place. Can't say you get a refund or not
without reading the repair orders.

The standard to have LEGAL recourse is if the diagnosis was reasonable. It is actually COMMON for there to be more than one possible cause for a problem. That said, I don't see have ANY gasket could be at fault for a squeal. Depending on what you actually told them before the diagnosis, it is possible that the exhaust manifold gasket actually needs replaced, but wasn't the source of the squeal. If you mentioned the squeal to the second shop, they would likely have started with likely causes of squeals, and never even looked at the exhaust manifold.