On the legal point, I'd be wary of some of the certainty being expressed on Mumsnet.
If the tree belonged to the neighbouring property owner, then there are several possibilities:
If the tree was healthy and simply came down in a storm, the neighbour/landlord may not be liable for consequential losses such as missed work or damage.
If the tree was obviously diseased, dead or dangerous and the owner failed to deal with it, they may potentially be liable because negligence could be involved.
Regardless of liability, it seems odd for the owner of the tree to take the position that they have absolutely nothing to do with a large tree that originated from their land and is now obstructing access.
The practical problem for the OP is that if they have no home insurance with legal expenses cover, they may struggle to get quick assistance. Insurers often end up being the ones who sort out responsibility arguments and organise contractors.
I'd also want to know who owns the shared drive. If the drive is jointly owned, subject to rights of way, or part of a management arrangement, that can affect who is expected to maintain and clear obstructions.
As for the tenants, they're not necessarily obliged to hand over the landlord's personal contact details because of data protection considerations. However, if they're renting, they should know where notices can be served or who the managing agent is. Under section 1 of the Landlord and Tenant Act 1985, a tenant can require the landlord's name and address from the person who collects the rent. That doesn't directly help a neighbour, but it shows there is usually a route to identifying the landlord.
My instinct is that the OP's strongest immediate steps are:
Check the Land Registry.
Contact the local council if a highway or emergency access is affected.
Check whether there is any buildings/home insurance cover.
Photograph everything before any clearance work is done.
And no, I wouldn't advise moving the tree into the road. That's one of those ideas that sounds satisfying on a forum but could leave the OP responsible for creating a highway obstruction.