According to Simon Lavington's "Early British Computers" (Digital Press (speaking of historic names) 1980) The LEO design was derived from EDSAC experience and discussions with Maurice Wilkes.
The purpose mentioned in the book was "automating office procedures such as payroll processing..."
This makes sense, and fits with a chronicle from Raymond Thompson (included in the book) indicating that a payroll program was run in January of 1953.
BEGIN CONJECTURE:
It is possible, and perhaps even likely, that the machine was also used in inventory management or in reconciling accounts and invoices. This latter task was a huge part of the target market for mid 1950's IBM machines. For many companies inventory and invoices needed to be reconciled by the end of each work day. The process involved something like "index cards" with what we would now see as database transactions (add X to account Y, debit Z from account W). Lyons managed tea shops all over the UK. Given that some of the inventory was subject to spoilage, maintaining inventory and monitoring demand in each shop must have been as difficult as it was valuable.
This notion is supported, perhaps, by the last entry in the chronicle for November 1954 (page 73 in the book)
Teashops General Report for Wembley teashops has entry: "The head staff at this shop would like to give thanks for LEO. This is a wonderful time saver, and work saver, and we are grateful for it."
Perhaps they were talking exclusively about payroll, but I think it goes beyond that. (Though historical speculation is ...)
Now that I've dug a little deeper this page
https://warwick.ac.uk/services/library/mrc/archives_online/digital/leo/story suggests that at least one application involved the valuation of bakery production.
END CONJECTURE;
In any case, Lyons was big business. LEO eventually merged with English Electric, and its market extended well beyond the Lyons company.