By way of conclusion, I report on the washing of the cream floormats. Both mats were in the same condition - 2 years' wear, although having been vacuum cleaned fairly frequently.
In view of the range of advice which Members kindly offered, I tried both methods: carpet shampoo diluted foam applied with a sponge, and vaccuumed when dried; and the same application (by sponge) of the carpet shampoo foam, immediately blasted off by power washer spray.
Obviously, the first method was much quicker overall because it took far less time to dry. The second method took 2 days to dry - one hanging on the washing line, and then in the airing cupboard.
Verdict: method 2 wins by a considerable margin! Much cleaner. And no ill effects for the mat.
Not surprising, really, given that the majority recommended it as the way to clean the mats.
Many thanks to you all.