It's amazing that they could put on a religious service each day in only 15 minutes, but what did it look like?
The BBC has an online archive of its magazine Radio Times and I looked at each day in January and July 1941. Despite the name, the Daily Service only took place from Monday to Saturday, but interestingly the book seems to have been used day by day in order, skipping over the Sundays. This fits with the goal, stated in the preface of Each Returning Day to provide a comprehensive arc of prayer each month, rather than in each service. A spot check suggest that some Services may have been omitted in order to keep the book and calendar it sync.
But since the description of each service was only a book page listing in New Every Morning and Each Returning Day we'll have to look at contemporary accounts and other broadcast services, mainly the longer Sunday service and services for school children. It was, in essence, morning prayer, with an abbreviated psalmody, and no sermon.
The Sunday services did not have a standard format, but -- in 1941 at least -- oscillated between "high" and "broad" forms, appealing to an all-out audience, both those in the established churches in England and Scotland, and Dissenting churches, Catholics excluded. So it's possible there was a standard set of texts with elements in common, but not a standard service.
But my interest isn't re-enacting those services, but seeing how that approach might make Sunday worship easier to plan, and the conduct of worship easier to teach. For next time.