Celtic has an interesting aspect within the European Indo-European context. Many similarities were found between the Celtic languages and Semitic (and Afro-Asiatic) languages, see this simple youtube report. In 1993 Orin Gensler did a PhD study at Berkeley on an evaluation on the Celtic/Hamito-Semitic parallels. He concluded that the similarities can not be a statistical coïncidence. While these conclusions have not been rejected, an acceptable explanation was never found. A theory by professor in historical linguistics Theo Vennemann (Atlantic languages) was rejected by other scientists. Vennemann had three separate explanations for Basque, Celtic and Punic for substratum and superstratum languages.
Now that we know how the arrival of the Indo-European languages in Europe is related to the migration of the R1b-M269 and R1a Y-DNA branches, it is usefull to see if these characteristics could fit in the knowledge of the migrations and the arrival of the Celtic language in Europe. We now know the majority of male line migrations and population growths.
The present males in Europe are related to the descendants of the arriving Indo-European language people. We also know that this is not the case for the female lines. In Germany and Scandinavia the majority of the autosomal (mix of male and female lines) is from the Indo-European ("Yamna"), but further south (Spain, France) and west (Britain) the majority of the autosomal is from older European lines. This fits with other information of the Indo-Europeans: they arrived in the Germanic area, and as they spread west and south, they mixed with larger percentages of European women. In the Iberian peninsula the Indo-European ("Yamna") autosomal is less than 25 percent, while the male lines were almost 100 percent Indo-European (R1b-P312) after the arrival of the Indo-Europeans, and before the arrival of the Phoenicians. This means that the female lines are in large majority from European descent.
If we correlate the Celtic languages with Y-DNA we see that the languages roughly correspond to the region which shares the R1b-P312 haplogroup.
Since the Celtic languages share common characteristics that are not found in the other Indo-European language, it suggests that these Celtic language characteristics had an impact on a large region. The impact could have been in the early period of the R1b-P312. That could be the case if the language of the local females spoke the same local language, and their children distributed the new proto-Celtic language structure. A second option could be that the Celtic language structure was present in the population of a large area in Europe and the impact came slowly. In this second option, it would mean that this language structure change would be the same in the different areas. This would mean that the structure was the same and the effect in a large geographic region was the same. Given the strong presence of I2 in this part of Europe (with in majority a shared ancestor less than 10000 years ago), it is possible that they spoke the same language structure in Europe. In this second option one would expect similar languages and dialects in all over the regions of the I2-tribes/Celtic languages. This second option does not fit easy with a language isolate language in the Basque language, which was in the middle of the I2-tribal regioni, but has a different structure than the Celtic language. The option where the Celtic language structure was present in the early R1b-P312 tribe before the spread of the language to a large region in Europe is independent of different local languages.
The easiest scenario has an ancestor of Indo-European male R1b-P312 mixed with local European women. Their language mix has the original phonology and structure of European women (who spoke an old European language) and the words of the Indo-European men. The mix was taught to the children and distributed of the R1b-P312 region of Europe. This mixing of language and the male and female parts fits with known knowledge of Language contact, Creolization and Genetic Linguistics (Thomason, Grey and Kaufmann, 1988).
This scenario does not mean that the people in Europe spoke an Afro-Asiatic language before the Indo-Europeans arrived. It means that they shared several aspects with Afro-Asiatic which were not shared with Indo-Europeans. Whether this could have been related to the language that was spoken in Anatolia before they spoke Indo-European, is guessing at the moment, but seems unlikely. Given the geographic distance and Y-DNA differences between I2 and G, one would not expect a shared language between I2 and G. If the mother language would be related to the mothers in the original tribes of Anatolia, one would expect similarities with Hattic which is thought to have characteristics of languages spoken in Anatolia before the Indo-European language arrived in Anatolia.
This picture (from eupedia.com) shows the difference in autosomal contribution in different areas in Europe. It corresponds to a larger autosomal ("Yamna") percentage in the R1b-U106 and R1a-Z283 region and a smaller autosomal ("Yamna") percentage in the R1b-P312 region.