Worried that my house and garden are sinking

KJIWTIStructE

750mm is the minimum foundation depth for a low plasticity clay, this means the clay soils will shrink and swell but to a lesser extent than other types. So assuming it is on clay then yes, reducing tree heights will lower the water demand and also with sustained rainfall the clay is liable to heave. If the trees are approx. 25m (80ft) then they are at mature height pretty much. Their water demand will depend on the species, some take huge volumes of water from the ground  others less. Cutting back trees is a remedy for subsidence but causes the opposite action as the ground recovers, this can take a long time, some years sometimes. If you have no visible cracking in the outer brick leaf then there is no immediate indication that you have foundation movement.

KJIWTIStructE

http://nhbccampaigns.co.uk/landingpages/techzone/previous_versions/2011/Part4/section2/appendix.htm

KJIWTIStructE

The link I've posted gives you a guide on how deep foundations need to be in the vicinity of different tree species in different soil types. If we can establish the following then we can work out if the foundations are deep enough.
1. Soil type, this can only be a best guess based on the bgs website.
2. Tree species, there are apps to help with this, use leaf or bark and follow the steps, if they are spindly but very tall could they be poplar? Maybe Google search of a few tree types can help you with this.
3. Distance from structure, this is the easy one that you only need a tape for!
We could then make an educated guess on how deep the foundations should have been if designed for the trees that ended up on this section of land nearby.