I would suggest posting on the Garden Law forum
Here. They were very helpful regarding issues that my parents had regarding trees.
From what I gather, now that they have put you on notice then you have to take responsibility to make sure the tree does no more damage to there property.
I
suggest that you go see the CAB or use any legal cover as part of your house insurance to deal with the issue. If you don't have legal cover then consider a solicitor if cab are not helpful, but they normally are quite good.
But
what I would do is contact whoever sent you the letter and ask them for proof that the tree in your garden caused the damage. No proof = no claim as you had no malicious intent for the tree to land on there property, which is the only way they can win in court.
If you want to be extra sure ( and the tree hasn't been trimmed since 3 years ago), then you could employ a tree surgeon to come and asses the tree to see if a branch is missing. ( They will be able to tell you and also advise on any future maintenance now that you are on notice).
I feel that they are trying it on as the garage is a very long way from the tree. I would even go as so far as email/ recorded delivery a letter to them with a printout of the image that you have, asking them how the tree was able to travel that far.
We have had very large trees fall from a neighboring churchyard in high winds and all it did was go straight down. It destroyed a dry stone wall below it, but didn't go sideways over 30 feet.
Unless the tree in your garden is much bigger than the rest of the tree's near it ( which it doesn't look like) then suggest to them that they visit the site before contacting you again.
Please don't take the above as legal advice ( as its not) I would still check on Garden Law as some of them are lawyers and specialists, and may be able to offer more help/advice.
Jon