I was talking about looking into 1-year MPH programs in another thread where I was asking my own questions (the one no one is replying to 🙁)
I spent a lot of time going through the list of accredited schools put out by the ASPH and CEPH and most of the 1-year MPH programs I can find are at the competitive schools: Harvard, Hopkins, Yale, Columbia, Berkeley, also Dartmouth. I found other schools where the "normal" MPH curriculum is more than a year but they say you can do it one year under certain circumstances and such: Univ of Texas, Univ Southern Cali, UC Davis. Tufts says their global health MPH can be done in 14 months. U Illinois Chicago has a "professional enhancement program" where I think you can do the MPH in 1 yr if you already have another related post-grad degree.
I can't open Dartmouth's MPH link either these days but it says elsewhere on the website that MPH apps are accepted on a rolling basis.