The most generic answer which avoids being obviously wrong by making specific claims is that IE Bell Beakers spoke a dialectal form of late, nuclear PIE that branched out of the Corded Ware Indo-European dialect. Where we go from here is where everyone disagrees.
Many say there was an Italo-Celtic phase. This may be necessary in order to explain similarities between Italic and Celtic, but IEs kept moving west all the way to the Atlantic coast. The biggest issue that leaves what precise language they spoke open to debate is the long linguistic dark age where we have no writings, not even inscriptions. Did Western IE Bell Beakers speak some form of Celtic then? To a great extent, the answer to this question may involve semantics over the definition of "Celtic". One objection is that the birth of Proto-Celtic is often said to be quite late. Two rebuttals may be given:
1. How can we be confident of a strict date range for a reconstructed language without written records?
2. If it wasn't Proto-Celtic specifically, then why not "pre-Proto-Celtic"? This raises the question of what we even mean by "Proto-Celtic", because if pre-Proto-Celtic is the direct ancestor of Proto-Celtic and nothing else, why not just call it Proto-Celtic?
Some insist that the original center of Celtic culture was Hallstatt, and Celtic languages radiated out from there. This idea isn't sufficient to explain all Celtic languages. Central and Eastern continental Celtic culture can explain Gaulish and Brythonic, since the evidence suggests a late Brythonic colonization of Britain. It cannot explain, however, Goidelic and Celtiberian. Without genetic studies indicating late population movements which would have brought continental Celtic speech, we may be forced to consider that Celtic languages developed naturally out of whatever IE dialect Bell Beakers spoke along the Atlantic coast. Evidence of linguistic colonization is necessary to refute this idea. Celtic denialism isn't an argument.