Celtic tribes in Europe were established over a very large area, from iberia to the British Isles to anatolia. Would they have shared a language? Or would they have very different languages? Would the celts in gaul have been able to understand and speak to one another without difficulties?
Proto-Celtic is based on reconstructions by common words in Celtic languages, but whenever a type of "Proto-Celtic" was spoken was the last time that Celtic language speakers spoke the same language. So, the short answer is that at one point, there was a predecessor to the Celtic languages.
However by the time that the Celtic languages had spread out, the languages had broken apart to distinctive language groups. Such as Celtiberian (Northern Iberia mostly), Gaulish (more widely spoken, from France to Turkey, though there still would have been dialectical variations), Brythonic (Britain), Goidelic (Ireland, later making its way into Western Scotland).
There was also Lepontic, which is the earliest attested Celtic language and was spoken in Southern Switzerland and Northeastern Italy. Noric was spoken in Austria, but may have been related to Gaulish. So with all of these different branches of Celtic languages, at their maximum spread, they would not have had mutual intelligibility.
/u/Sirovoltos provides the answer in a succinct way. To that I add that the question would have been better presented as being about "Celtic-speaking people" and "Celtic-speaking tribes." It is very likely that what spread was a language more than a people. While some people may have been on the move, it also seems that in many situations Celtic languages became a useful means of talking with neighbors and it consequently diffused and was accepted as the preferred language in many places. This seems to have happened independent of actual movement of people (although some of that likely occurred as well - genetic material has a way of moving from place to place!).