No issue parking in Tewkesbury, but Worcester can be a problem. A bit of a climb from both nearby car parks. I cannot manage it using my elbow crutches. Even with a mobility scooter the one car park near to the cathedral you may struggle. Worcester old town is pleasant to visit and if you know where to go parking should not be a problem with a BB.
At Gloucester docks there is a huge car park so no issue however I cannot recall if there is a lift in the building where the museum is located as the last time we visited I never had mobility issues. That museum is well worth a visit. Also nearby is the Gloucester Regiment museum.