Not from my experience....
I broke down with a bust water pipe (different one to the other time!) and an AA van pulled into the garage i had managed to get to. I asked him to have a quick look and offered to pay to join if he could sort it. He said even if i joined on the spot he couldnt fix it as he didnt have the part, and wouldnt recover me due to the 24hr thing.
I ended up having to pay 80 quid to get a local recovery firm to get me to a vauxhall garage where i had to wait till the morning till they opened