Can you elaborate a bit on why you think so? The model presented by Chang et al. has estimates a bit on the lower end of the time spans I have seen for IE splits compared to other sources I've had access to. IIRC some of them assumed Germanic and Italo-Celtic for instance to have diverged from other branches even before 3000 BC, so in Early Yamna times. Probably a middle ground is closer to the truth. But in any case I did not get why you think a BB = NWIE would require the PIE breakup to be as early as the Neolithic (mind you, I do not believe undivided PIE was spoken in Yamnaya at all, I think that, if Anatolian and even Tocharian are included, the split started even before the Yamnaya expansion, and by the time of CWC it was already split into very close but distinct sister languages, with Proto-Anatolian probably already well developed).
As for NWIE, if Proto-Celtic and Proto-Italic started to break up roughly in 1200-1100 BC, we can assume that the languages had started to be spoken, breaking from their common ancestor, at least ~600-800 years before, so roughly by 2000-1800 BC. And if you add Germanic - if Germanic does indeed derive from a common ancestor with Italo-Celtic, which is not totally certain AFAIK - that date must go back several centuries further, so roughly to 3000-2500 BC (on the lower end coinciding with the begining of BBC in Central Europe). And of course we do not need to presume that NWIE started to be spoken only when the Central European BB phenomenon got formed. The language was brought with the newcomers from elsewhere and it didn't start from scratch straight from Undivided PIE. The timing in my opinion is totally compatible with a NWIE dialect (not necessarily the only one that existed, but the one that was successful in the long term) being spread by Rhine BB people to many areas west of Germany (though not necessarily all, as we all know the BBC had a very sparse occupation of territory, and they may have possibly adopted the local languages in some of the places they settled in, as in the hypothetical case of the Basques).
As for no IE language with such time depth existing in Western Europe, the vast majority of it had no written language virtually until the Common Era. The areas that had written inscriptions from earlier on had writing only during the later Iron Age and were basically those most impacted by Celtic and Italic conquests. That situation is not very conducive to allow us to see remants of earlier NWIE languages (though Germanic, I insist, is definitely one such example, it's definitely divergent and innovative in comparison to Italo-Celtic). I also have to doubt that had the all-important BBC been associated with another language family, we wouldn't see the non-IE remnants of that coherent and homogeneous language family spread to many parts of Western/Central Europe. Instead, we see Basque and Iberian, possibly but not certainly related, Tartessian, Etruscan/Rhaetic (Tyrrhenian), different language stocks, and not a formerly powerful and expansionist language family competing with IE.