Will the shoots coming from the root of a dead ornamental flowering cherry eventually grow into trees?

They might, but it depends whether your tree was grafted onto a rootstock as to quite what you'll end up with. Some Prunus are grafted onto a more vigorous rootstock, and that might be what you've got growing.

Many thanks for your swift reply! The tree was a pink flowering cherry with a white flowering cherry grafted onto it; so we had a sort of combination tree. But I suppose the original pink tree could have also been a graft onto something else? Have you any idea what type of rootstock is usually used? I have potted one up and left 2 others growing, but if it is likely to be very vigorous, I will have to remove them as they are quite close to the house. Any idea how long before they flower? (Sorry for so many questions ; I am quite good with flowers and veg, but I am a bit of a novice with trees!) S++

If the whole thing was grafted onto a vigorous rootstock, then it will be just that - very vigorous, compared to an ordinary cherry. You should, though, be able to see a graft point or lump, below the graft for the twin varieties, unless you've already sawn it off past that point!
If it was me, I'd be poisoning the woody roots and the stump, but that's just my preference.