Sarcoptic mange (scabies) is a parasitic disease which may be contagious. That said, not every dog is equally susceptible. If your English Bulldog truly has sarcoptic mange (as opposed to demodectic mange) there are several medications and protocols for treatment. The good news is that sarcoptic mange is virtually always curable, unlike demodectic mange. I would simply keep the dogs apart, feed and handle the well dog first, wash hands, and dont share implements such as combs or bowls. Sarcoptic mange mites do not survive long in the environment. Keeping the dogs "biologically" apart will likely be enough to prevent transmission.
Do confirm that it is sarcoptic mange by having skin scrapes done by your veterinarian. If it turns out it is demodectic mange instead, then your other dog will not get that, as it is not contagious, however it is much harder to treat and may be refractory to treatment if widespread. I add this because Engish bulldogs are a breed not uncommonly affected by demodectic mange.
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