Neither midst nor railway is all that unusual for American speakers.  “I’m in the midst of something” is a perfectly normal expression. (BTW, the d in midst is often unpronounced, and, in student writing, is sometimes not spelled or typed, which leads to delightful double meanings: “I’m in the mist of problems.” Railroad might be preferred, but railway would not sound at all odd.  One of America’s largest trucking companies is, or used to be, Railway Express (I don’t know its current status).  Perhaps the preference for railroad is that railroad can be used as a verb, meaning “to hasten to a conclusion; or to devise false evidence against,” but railway can’t have this usage.