How were the letters written on the same day in the Guernsey Literary & Potato Peel Pie Society delivered so quickly?
The Guernsey Literary and Potato Peel Pie Society by Mary Ann Shaffer and Annie Barrows is written, as you suggest in your question, as a series of letters. Many of them are letters which are sent between Juliet Ashton in London and citizens of the Channel Islands, of which Guernsey was one. These would have gone back and forth regularly on packet boats, and the dates of these letters account for that. The same is true of the letters between Sidney Stark, who is in Australia, Sophie, who is in Scotland, and Ashton in London. There are also telegrams which, of course, would have been sent and received quickly.
The letters to which you refer are actually more like notes, and they are sent between Markham Renolds and Ashton, both living in London. This novel is, of course, a work of fiction; however, it is likely these notes were delivered by messengers hired by both parties. It is a system no longer used in most places, but cities like New York do still have a semblance of this practice in the form of bicycle couriers. Simple notes of invitation, for example, are generally sent electronically; but documents or contracts or blueprints, among other things, are often sent via these couriers. Post-war London undoubtedly had such messengers, especially during the rebuilding process. (Remember that Ashton's phone was buried amid the rubble of her apartment, and she was certainly not alone in that circumstance.)