Hi T-Tree,
Clay soil shrinks as it dries out. There's what you see at the surface, which is a big swing between summer and winter, but often clay soil can be very deep and it's the overall moisture content of that which matters. The surface will be muddy in the winter and rock hard in the summer, but deeper down and throughout the whole 'layer' this variability is far less.
What does happen is that where land is built on, the amount of water that reaches the clay is generally reduced - houses have roofs and hard landscaped area (paths, driveways, etc), which takes water which would have previously gone into the soil away. So overall, it's fairly normal for clay land that's been built on to shrink. It's sort of pot luck but generally, you'll see a bit of settling in a building on a clay soil. This is the lintel cracks I mentioned.
Houses have gardens and people plant trees in gardens to make them look nice, amenity, fruit, etc... so after the building hapens, there's then a period where people plant trees and patches of this land where those trees go are subject to more drying - especially with very vigorous, quick growing trees, the area around a tree will become drier and therefore the soild around the roots will tend to shrink more, and more quickly.
For example, I have a fairly mature Laylandii near the back corner of my bungalow. We live on a flood plain on deep clay. That corner of the bungalow has a big-ish crack, not a wide crack, but nevertheless noticable. We asked our builder and he says that the rule of thumb is, if you can't get your fingers in the crack, there's nothing to worry about! Not sure about that, but he knows more than me. The ground under that corner has shrunk a bit more than the rest and that back corner of the building has dropped a bit... enough to open up a crack. We have a bit of very mild cracking elsewhere in the property, but this corner is worse.
If we remove this tree, the bit of garden where it was will become 'wetter' again... so the soil where the roots are will expand. Proably all that will happen is that the crack will close up again, because the tree came after the house.
It's quite a complicated subject and if you have any major concerns you should speak to a structural engineer... but if you have a house which is built amongst existing trees, on clay soil, and you remove these trees the soil will become wetter and expand. This is 'heave'.
Say you house is completely surrounded by trees, and they were mature when you built the house, removing them would make the soil all around your house expand and probably you'd not notice anything. It would 'heave evenly'. Maybe. If you had the same scenario but say two big trees at one corner and you removed them, the soil there would expand and that corner of your house would be lifted and... you'd probably get cracking, or worse.
With the original poster, I'd be concerned about that conservatory - if it was added on after the house was built, and that big old tree is removed, and they're on clay, there might be some heave! Probably the house will be fine... but it might have a bit of cracking already.
Imagine you've got a big gingerbread house and you try and pick it up... if you can get a tray under it you'll probably be fine, if you get your fingers under one corner, it's going to end in tears!!