Upvote:3
On multiple occasions, I have found that taxi offices near train stations have such a service. I have done this both in Exeter and in Penzance when we wanted to explore the town or surrounding area, but without our luggage. They charge a fee for the service. You might have to wait a little for someone to be available, and make sure to make an appointment such that you don't come in front of a closed office 10 minutes before your train departs.
Upvote:5
In your specific case (Peterborough), I cannot find any official luggage storage facility. You might get lucky with the museum or one of the city centre hotels, even though they don't normally offer this service.
I'd suggest presenting yourself as a genuine museum visitor, hotel restaurant diner etc. and being as charming as possible! For example, arrive at the museum, buy a guidebook and engage the staff in conversation about the awfulness/loveliness of the weather. Then ask if there's anywhere you could leave your case while you go round the museum. If you get the right person on the right day, you might find someone who will bend the rules, but be understanding that they may be forbidden from storing luggage for security reasons. The more genuine you appear, the more likely you will be to get away with it.
As a more general answer, I would first approach the Tourist Information Centre for that town, if one exists. Some TICs have their own left luggage facility (e.g. Bristol, Greenwich) and others will know of e.g. local museums which have a left luggage facility. There is also a UK-wide service called CityStasher which operates in a number of cities.