If I wished to place MWW in the fifteenth century (which I don't), it would clearly have to go before 1 Henry IV, since there's certainly no time for the events to have taken place later than that. I'd have to go back to the play to see whether this is technically possible, but I'm not going to do so because I've no interest in making that case. I agree that it clangs.

It clangs no less resoundingly for me as a sixteenth-century play, because of all those fifteenth-century figures in it, several of whom are strongly associated in my mind - and, I suggest, that of Shakespeare's audience - with important historical events such as Agincourt (and one perhaps with the historical Oldcastle). You may feel that the audience would have been able to disregard these elements to a lesser or greater extent, but off hand I can't think of a single other play of the period (by Shakespeare or anyone else) where this kind of unremarked transplantation takes place - except where figures such as Gower or Machiavelli are used as Prologues, but there they are clearly outside the temporal frame of the plot. I don't think we can claim it as a convention.

So, I agree that it doesn't fit anywhere - which is perhaps a better way of putting it than to say it's ambiguous (which may imply that it fits in more than one place), and that in that sense it may indeed be called timeless. In my opinion, this is sufficiently different from its being "a play set in contemporary England" to justify what I said to vschanoes - where I was distinguishing it from plays such as A Chaste Maid in Cheapside and The Alchemist, which clearly are set in contemporary England.