Eventually I worked out why my patchwork pieces didn't fit. The problem was not in my cutting but in my sewing. I used the 2 mm mark on my sewing machine foot as a guide to the width of the seams, whereas I should've used the 5 mm mark!
Salvage job required. I removed the outer row that I had cut to new measurements because it made the piece too big for a tray cloth and didn't look right. Then I added a backing and did the binding as per the original instructions: