The best podcast app that I have used (and I have used a fair few on Android and iOS) is Pocket Casts.
I don't know about button presses - there are views you could choose (such as "downloaded") where you would just get a chronological list of your podcasts. SO - it would only be one press from there to play.
Even from the main view it's just 2 presses to play your podcast (one for the podcast itself, one to select that episode.)