A while ago myself and the wife had a rainy week camping in the Tewkesbury region in a tent, but more importantly with our paddleboards. We stayed at a campsite to the side of the Lower Lode Inn which is great because of course it's a pub, but also because there's a slipway and jetty for launching from. Food was decent and they have Doom Bar on tap.
This gives you access to the River Severn, and across the water to the bottom of the River Avon. We chose to do a leisurely paddle up the Avon as we had the dog with us, and to try filming on a borrowed action camera. Luckily we got two hours of sunshine all week, one of which was captured in the footage.
The river is pretty scenic, and a nice quiet paddle. That the top of the river is a weir next to Tewkesbury Mill, which I don't think you can get around so limits you to this really easy paddle. We'd planned to head up the Severn, but weather was rubbish so we moved onto the next location, but this would provide a longer paddle as I believe you can get all the way into town. Not tried it though so I might be wrong.
All in all, a really nice location for easy paddling in Gloucester. The pub is popular with kayakers in the summer so if you're staying there you might want to book ahead.