Very nice hypothesis, I find it reasonable. You are right that I (we) should be more prudent dealing with all these reconstructions. However, we need to fine tune the chronology here to see if this scenario is really plausible. If the Italic tribes and the Celtic tribes were still all around Hungary way after the breakup of Proto-Celtic and Proto-Italic, and well into the development of such distinct languages as Oscan and Gaulish, that may make sense. Otherwise, I don't think we could explain well why early Italic and early Celtic became "P-shifted" in some places and not in other places, if they were supposedly under the same areal influence or, perhaps, the same elite dominance. Proto-Celtic and Proto-Italic do not show any kw > p, so we can't assume this is an old sound change, before the widespread Celtic and Italic migrations to places far away from Hungary or its surroundings - unless, of course, we find evidences for it.
Since Celtiberian and Lusitanian, which are plausibly more conservative, "archaic" remnants of early Proto-Celtic migrations, also lack the kw > p change, I'd speculate that this is a latter change, from the times when Celtic (and possibly also Italic) languages were already dispersed in a very large territory, where the "Hungarian" influence would've been less strongly felt. But, anyway, perhaps just some branches of Celts and Italics, already divided and dispersed, were under this influence that led from *kw to *p.
However, in any case, this must have very little or nothing to do with Greek "íkkos" > "ippos". This change happened nowhere near Hungary, but mostly in Attic Greek, and in an entirely different cultural and social environment. Also, the change was completely different in nature from the Celtic and Italic sound changes, since the channge to "p" only happened in some words under very specific conditions (before [a] and [o]). In other situations, the changes were very different, from *kw to [t] or [k]. It was apparently an independent process.