The 'chestnut tree' is probably a horse-chestnut (Aesculus hippocastanum), rather than a true chestnut tree belonging to the genus Castanea. The horse-chestnut is a large deciduous tree which was introduced to Britain in the 17th century and became popular for its attractive blossom in springtime. In autumn its nuts, known as conkers, have traditionally been collected by children to play a game of the same name.
There is something quintessentially English about the idea of 'tea under the chestnut tree', possibly because of its association with poetry such as Rupert Brooke's 'Stands the church clock at ten to three/And is there honey still for tea?'* or Longfellow's 'Under a spreading chestnut tree/The village smithy stands'**. It is, therefore, very much in keeping with all the other nostalgia-evoking concepts which come to the narrator's mind during her exile from England.
* from the poem 'The Old Vicarage, Grantchester'
** from the poem 'The Village Blacksmith'
Bearing in mind the parallels that have been drawn between Rebecca and Charlotte Brontë's Jane Eyre, it is also interesting to note that Rochester proposes to Jane by 'the giant horse-chestnut', which is later struck by lightning.