Good question, for showers are the main/sometimes only reason for going to a campsite.
And good to see some camping/wild camping stuff on here.
I remember that when I inter-railed as a youth, many nights not actually under a roof, a fair few main italian stations had showers - don't know if they still do. Never seen such a thing in my entire life in this country.
I fear though that in a country where even truck drivers, the backbone of the nation's distribution, can't find showers, the answer to your question isn't going to be very positive.
I'll just live in hope of at least sneaky suggestions for getting showered when freewheeling on the road in the UK.