Puddings refer to everything from treacle sponge to sorbet (don't know what a sorbet would be referred to if served between courses, but suspect Mitfords wouldn't have encountered such weird ways), dessert is fruit (eaten with appropriate cutlery). Just look pityingly at anyone who queries pronunciation and ask, quietly, as if something shameful, "Oh....do you not know about syntactic relativism then? Did you not..receive much of an education? Perhaps your parents didn't prioritise it?"
But back to Archers - Lily can't be that posh, or only shakily so. Her mum is the daughter of a farmer (and not even a mahoosive wealthy one like Brine) and a former product demonstrator, who married into a family on its knees and a breath away from losing the house which is its only claim to any kind of grandeur. Now it's hired out for weddings and decidedly non-U corporate events. Lily went to a shonky private day school, not even anywhere particularly posh. It's all very Normal for Norfolk, and not even as established as that.
So if she said sweets/dessert rather than pudding, no-one would be so surprised at her being slightly 'off'. (I do think this is all bollox, by the way, but you know - TA does rather invite this sort of tosh).