I refer to Mawer's PNs of Northumberland and Durham.
p.169 gives "Ruchester (Chollerton) 1348 [...] Rowchestre. [...] the first element is the adj. "rough""
p.170 gives "Rudchester (Ovingham) [... numerous forms in Ru-, Roue-, Rou- ...] possibly Red-one's chester".
1. The modern map has no Ruchester. There is a Rouchester Farm at NY 891 775, about 8km NW of Chollerton. Did Mawer mean this place?
2. Why does Mawer make no link between the two names? The simplest hypothesis is that Rouchester Farm was named from Rudchester, since it is very unlikely that there was a Roman fort at Rouchester.
3. Why isn't "rough" a likely origin for Rudchester (NZ 113 693)? The occasional -d- could be an influence from "red" (Whitchester is close by at NZ 098 683).