Hi.
Thanks for the extra information about the insurer's loss adjustor's report.
It seems that the foundations are indeed rather too shallow if the report is to be believed. For a willow 6m away, NHBC would suggest foundations 2.5-3m deep depending on the particular subsoil!! Obviously 300-350mm is not enough to get below the zone of influence of the roots - which is presumably why your house is now moving. In fact 750-1000mm is normally the minimum required by NHBC/Building Regulations in a shrinkable soil so the builder has fallen some way short of that too.
So firstly, the report maybe warrants some questioning:
"Boring holes" is a particularly rubbish way of establishing the depth of existing foundations. You actually need a proper trial pit - at least 1mx1m in plan, taken down to 200mm below the underside of the existing footing and taken slightly underneath to confirm that it is genuinely the foundation base. My first line of attack would be to go back to the insurers and put this to them. Say that if they are refusing cover they need to prove their claims conclusively by sight - not hints from an unreliable borehole. You also want root samples to be taken and sent to Richardsons (at Reading Uni) for identification to prove which tree is responsible. Sadly this does mean that someone will need to dig a big hole in your garden though.
Secondly to prove subsidence you need to monitor the cracks for at least 12 months so that you can show a cycle of shrinking and swelling through the seasons. Ask that they do this now that they have installed the measuring tags. What type of thing has been installed? You say "dots" - are these little metal studs with dimples in the centre? These are DMec studs and if they are either side of the cracks you can buy a device to read them monthly yourself - if it comes to that! It might be that the movement is not tree related (although it sounds like it might be).
Thirdly in relation to the inferior workmanship - My first thought on this is that it is a failure of design/specification rather than workmanship - ie choosing the wrong depth for the foundation, rather than constructing it badly. So to claim against someone you would need to find out who was responsible for the decision and then find out if they have professional indemnity (PI) insurance. It all sounds like a complete nightmare and I don't rate your chances of success in this regard. I am not 100% sure that you would be able to initiate a claim on them (if you can find them) because they only have a contractual link with the original owner and not you. I think this link can be transfered via a collateral warranty but the chances are that you don't have one. So this is another question for the solicitor probably.
Fourth - If the tree has to stay, is there any chance you can find out from the neighbours when they planted their tree? Because a tree can grow considerably in 10 years and there is a chance that it was not there or too small to see over the fence when the extension was built. This would show that the original builder/designer had acted reasonably (although to be honest this can only be said if the foundations are 750-1000mm deep) and it is actually the neighbours who have created the problem by planting the tree. This is a bit of a get out clause for your insurer as I think it means that they can claim against the neighbour's policy - although I suspect that your neighbours would not end up too happy about this and it would affect their premiums etc.
Fifth - I agree that it is worth looking into the completion certificates and whether the checks were carried out properly - either to show the insurance company that you did proper due diligence or to start an argument with the solicitor for compensation....
All of the above will result in a good deal of angst and trouble, and may not result in the insurance cover being reinstated to the level where you can underpin the house. However an alternative would be to speak to your neighbours. Would they consider cutting the tree down if it can be proved by monitoring and samples that it is their tree causing the problems? I normally hate suggesting this option, but if you have no insurance cover it is probably the quickest (and only surefire) way of making the problem go away. I suppose that this will entirely depend on how well you get on with your neighbours and what if anything you can do for them to encourage them to remove the tree.....
Good luck and let us know how you get on!