In JA's era, tea was invariably served at the more splendid evening entertainments, whether concerts or assemblies (public dances.)
Tea was frequently included in the price of the admission ticket.
Tea was generally set out in a seperate rooom to which everyone eagerly trooped to during the interlude, hence the squash.

The New or Upper Assembly Rooms at Bath all owned a large stock of tea equipment i.e cups and pots.
Gentlemen were in charge of procuring tea for the ladies of their party, so Mrs Allan & Catherine probably felt rather awkard without Mr Allan.
When Catherine attends her first assembly at the Upper Rooms, she had no dancing partner to escort her to tea. She and Mrs Allan squeeze in at a table that is already occupied, only to find themselves tea-less.
'Here are no no tea things for us, you see'. Eventually a gentleman at the table takes pity of them and helps them to tea.

(From 'Tea with Jane Austen. Kim Wilson. 2004)

Possibly attendents came round later to pour more hot water and clear away used tea things.