Yes, that's certainly plausible. It might be a little on the late side but certainly within the realm of possibility -- these were languages that, for the most part, were only attested in writing very late, and only then it was many centuries after the root language would have been extinct. Some very educated guesses can be made about chronology and relationships based on comparative reconstruction, but the timeline is daunting enough in size that it makes it difficult to assign very specific dates to anything. Even the dates for PIE itself (which, due to the sheer number of daughter languages, is in some ways easier to reconstruct) are very broad.

But in 1250BC it would certainly be plausible to say that people in many parts of northern Europe would have spoken various forms of what's sometimes called "dialectical Indo-European" with what had to have been varying degrees of mutual intelligibility. On the other hand, they've just done some estimates of a massive battlefield around this time, indicating that people from a very wide geographical region were allied politically (in some way -- enough to fight, anyway) and obviously traveled around quite extensively.

But I digress. It's a decent guess, certainly serviceable for a work of fiction.

To add on, is there a more recent common ancestor to the Germanic and Celtic languages than Proto-Indo-European? The tree model often pushed would seem to say "no", but, being so close geographically, one would assume that Old Norse was at least a little bit more genealogically closely related to Gaulish than it was to Sanskrit.

What was the title of the book? It sounds interesting..

Not really. There were a lot of unstable features in Proto-Indo-European that resolved themselves really quickly and very differently in the Indo-European languages. Specifically with regards to laryngeals and syllabic resonants. In this regard, Celtic is more similar to Italic in how it dealt with both, and Germanic is more similar to Balto-Slavic in how it dealt with both. This definitely pushes the date back for common descent. Given that Proto-Celtic is roughly dated to 800 BC, and Italic is a little older, it seems very unlikely.

The pre-Germanic and pre-Celtic dialects might have had some interaction, but not much, as Germanic was relatively late to the central Europe game. Most Celtic loans are late and very much identifiable by unique Celtic sound changes: pre-Celtic *plowdʰom > Proto-Celtic *ɸlowdom > late-Celtic/pre-Germanic *lowdom > Proto-Germanic *laudą > English 'lead (metal)'. Compare the Latin cognate 'plumbum'. Note: both the Italic and Celtic forms seem to be borrowed from a non-Indo-European source, as they show difference in realization and a phonotactic feature that Indo-European didn't tolerate.