Yes, similar experience with a surveyor not drawing our attention to some things he should have seen enough of to raise concerns about, concerning the overall roof structure and quality of work. So actually more indirect than your case, where he could have seen the asbestos itself.
We pursued the surveyor's complaints procedure initially, then were able to work with a reputable litigation solicitor on a no win no fee basis. At that stage it was him vs the surveyor's insurer. We settled before going to court for about £10k, which was the cost of the remedial work required, plus all our expenses were covered.
He'd refunded his fee early on, as a goodwill gesture - no admission of responsibility, accepted on a no renunciation of any claim basis, you need to be careful about that. But that fee is really not the issue. The question is what would the value of your house have been, had the asbestos been known about (depends on market conditions, inconvenience etc as well as cost of remedial work).
Surveyors are regulated and insured, so must have a complaints procedure and are in a position to pay up if found against.
First check your contract with the surveyor and the survey report. Should he have looked in the garage and did he? Did he do what his contract said he would? Was the asbestos visible to a trained eye (not covered).
You can start by raising any issues and following his complaints procedure himself.
Then there is RICS guidance on how surveys should be carried out and written. Ours was generally sparse and hadn't used a lot of the standard sub-headings, so just hadn't said anything about a lot of things, like bits of roof, where he should have at least have mentioned their existence. You probably need to talk to another surveyor to get an opinion on this. You can pay them for a brief visit and an initial opinion. If you go further you'll need to pay them a few hundred for a written report.
I spent a lot of time finding out who to phone and talking to them - specialist solicitors for an initial opinion, most will give you 20 mins on the phone free, to find out if you have a case - then surveyors. I must have talked to all the buildings surveyors in town (structural surveys are now called buildings surveys, it's a different professional qualification from being a general chartered surveyor, you must use someone of at least equivalent qualification to your original surveyor).
I found there was an established, reputable network of about five buildings surveyors in a largish town. They all knew each other and could recommend who would be best for my needs. But, many will not do work against another surveyor, even though ours was not in their 'group' (which was telling in itself), they just don't want the angst. One or two would though.
So, lots to do, many stages to go through, money to depend - once you're at a stage when a solicitor thinks you have a case. From what you've said, you could well have a case but you need a professional surveyor's opinion on that.