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A friend has offered to dig up a water shoot growing up from near a greengage or apple tree. Can it become a new fruit tree and will it have roots already on it? Also, what is the effect of a water shoot on the main tree?

'm not sure thats a watershoot as such,sounds more lke an offshoot of the rootstock(whch aint much use to you unless you root t up and get some budding matierial from ABOVE the graft...... may be confusing what you're saying,a picture would help.

I think OJ's right, if you have to "dig" it up it must be coming out of the ground and therefore it's a sucker.
Suckers grow up from time to time from roots and it is of course the rootstock that's sending them up. You should pull them up, not cut, or eventually the shoots will become the main tree at the expense of the one you want.
If you cut them off they will just shoot again, only stronger, pulling them up leaves a jagged edge underground that will rot.

Water shoots are technically the strong straight shoots on a mature branch.

Can it be planted to create a new tree that will produce fruit?

Probably not.
It will produce a tree OK but it will be a nondescript wild type of tree and if it has any fruit at all it will take many years to start fruiting and will probably be as tasty as old boots.

What you could do is to grow the shoot and next winter graft a nice shoot from further up the original tree on it. Then you will have an exact copy of that original tree. It's not too difficult to do and you will have 6 months to read up on the process.

That sounds like a more interesting exercise. Thanks!

How long would it take for the grafted version to fruit?

Probably 2 to 3 years if you're lucky, certainly before 5 years anyway.
Depending on which part of the donor tree you take the scion shoot from you might get some flowers on the first year but they won't develop, but if they do you should pick them off to allow the tree to use all it's energy in new growth.