I am looking to buy a flowering cherry for a small back garden, soil is very dry due to next doors huge oak tree that takes moisture from our garden. The oak doesn’t cause shade to our garden as it’s position is on the north side of our garden.
You will need to improve the soil over a wider area than just the planting hole so that your new tree can get its roots in over winter and probably water it thru dry spells in the growing season until it's a few years old and well established.
You don't mention how small is small. One of the best understated flowering cherry trees has got to be Prunus x Subhirtella 'Autumnalis'. They may not have the rainbow autumn colour or the classic big blast of blossoms in spring, but makes up for its slow delicate flowers that stud the branches throughout the winter. Sometimes for almost 5 months through the bleakest and darkest days. A great tree to set off evergreen shrubs and spring bulbs.
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Prunus serrula has lovely, shiny, coppery bark which makes it a feature in winter. Good leaf colour in spring and autumn and small white flowers - https://www.rhs.org.uk/Plants/14025/Prunus-serrula/Details
If you want one with more vivid flowers - https://www.rhs.org.uk/Plants/99553/Prunus-Kanzan-(d)/Details
This one has purple foliage - https://www.rhs.org.uk/Plants/99810/Prunus-cerasifera-Pissardii/Details
Lots of choice depending on how blousy you want the flowers to be or whether you prefer a longer season of interest.