While many Hungarians don’t speak English, it’s still not too hard to find good English-speaking services such as doctors and dentists in Budapest.
Our doctor’s office is fantastic, and everyone there speaks English. Our kids’ pediatrician, for example, who is Hungarian, went to school in Chicago and doesn’t even have an accent. You can find the office here, if you’re interested: https://www.rmc.hu/en.
It’s the same with dentists. We have been to Helvetic Clinics (https://www.helvetic-clinics.eu/) a few times, and they all speak wonderful English. In fact, I think they get a lot of medical tourism for folks from the UK and elsewhere, as the services are cheaper in Hungary (roughly EUR 50 for a cleaning and maybe EUR 300 to have a couple of fillings done).
We don’t have close familiarity with the other services you mentioned (hairstylists, massage therapists, vets, etc.), but those should be easy to find if you look near the city core, where there are a lot of tourists. For example, anywhere near Vörösmarty tér or Deák Ferenc tér or the pedestrian/shopping street Váci utca, you should be able to find English-speaking services pretty easily. A few times, for example, we’ve gone into the Vodafone store near Deák Ferenc tér and have had English-language help. A great many of the younger people in town will speak English, since they’re being taught it in schools now. My wife went into the H&M once on Váci utca and was actually greeted in English upon walking in (quite a surprise for her, let me tell you!).
By the way, Váci utca is not too be confused with Váci ut, which is a street in a different part of town. Váci utca is near the river.