making a larger vegetable plot

Lift the turf now if you can and stack it upside down somewhere to turn into loam, dig over the area and mulch well with compost or well rotted manure.

If the tree is dying you would be better to have it removed, veg. wont grow well near it.

Now, if the weather is ok and the soil is not waterlogged, is as good a time as any. Leave the soil roughly dug to allow the winter rain (not frost as people suppose) to break it down ready for Spring. If you have humus of any kind, dig it in, but do not add either lime or fertiliser until Spring.
By sucker shoots do you mean growths from the roots or long thin whippy ones on the tree? If the former then you need to remove them as soon as possible back to the roots. I the latter (water shoots) then wait until Spring and prune them back to 2 or 3 leaves from the point of origin if you intend keeping the tree. You may not get any blossom on them, but this is how you turn them into fruiting spurs for following seasons.
Typed under difficult cat induced conditions.

Snap

This is as good a time of year as any for digging up the grass, Aina. After all the rain the ground will be nice and soft, so the job should be relatively easy. Pile the turves upside down in a neat block in a corner of the garden. Left like that they will make a good loam to be used in two or three years time. Then just dig over the bare earth roughly with a spade and leave it over winter. The frost will break down the lumps and you can start making a good bed for vegetables later in the winter.

Double your snap, Owdboggy!

At least we are all saying the same or similar things :-)

I must admit I always thought it was frost which broke up the soil, but modern research has shown it is actually rain which does it, otherwise areas which do not get frost would have lumpy soil in Spring.

lol

Well, we get plenty of both so should have good soil. ;-)