Hi Helen
Yes, you can cycle from the site and Blackmore and the surrounding areas are all flat so I would think it would be easy for you. Although, this isn't one of my strong subjects as we ourselves do not have bikes so I am probably not the best person to advise you of this.
The nearest village is Hanley Swan, and I would say it is about a mile walk. To be honest we have only ever been to the caravan accessories shop there, as we tend to drive out for the day. A lovely place to visit is Bourton on the Water, if it is just the two of you with no children then it is a lovely little village to go and have a pub lunch and sit outside if the weather is nice.
There is some lovely little village pubs in a little village called Guarlford, which is about a 5 minutes drive from Blackmore. From there, it will then take you into a little village called Barnards Green where they have a Somerfield, and a lovely fish and chip shop. They also have a little shop that sells the local cheeses and cider.
If you are driving then the postal code for your sat nav is WR14 3LY, that will direct you into Barnards Green from Blackmore and you will pass through Guarlford with all the little pubs on route.
Blackmore has a fish and chip shop on site every Saturday night, but queue early as they also queue up from the CCC site next door.
Nothing more to tell you, except I am so jealous and wish we were coming with you ;-).
You will have a lovely time, it is a really lovely site and my kids now think of it as there second home. They love it so much that we have even booked to go there for New Year.
Have a lovely time
Sam