In my opinion, most of the J1 in Europe is of Neolithic or Chalcolithic origin. The spread of J1 goes hand in hand with haplogroups E1b1b and T. All three haplogroups are common in the southern Levant, where agriculture arose, and would have spread via Anatolia, Greece and Italy, and by the intermediary of the numerous historical peoples from these areas. The Phoenicians, Etruscans, Greeks, Romans, Byzantines and Arabs all contributed to the further dispersal of J1, often reinforcing the frequency in regions already settled by Levantine/Anatolian people during the Neolithic. It is ludicrous to think that only one historical tribe (meaning post-Bronze Age, as historical implies that writing existed) is responsible for the presence of J1, E1b1b or T in any part of Europe.
In Iberia, J1 is obviously a cumulation of Neolithic, Phoenician and Arabic settlements. IMHO, the Neolithic farmers were the main contributors, because the hotspot of J1 in Portugal and Andalusia matches the hotspot of E1b1b, J2 and T, which means that the four of them came together. Considering the proportions, it rules out an Arabic or North African origin, and points rather at a Levantine/Anatolian origin. This hotspot also matches the earliest known Neolithic sites in Iberia.
Likewise J1 in Central France and Germany is most probably of Neolithic origin, because the Arabs, Phoenicians and other historical Middle Eastern people didn't settle the region. Only the Romans might have contributed to slightly higher levels in France - but they obviously didn't settled North Germany and Poland.
The J1 hotspot in Bosnia is almost certainly due to the colonisation of the region by Muslims from Turkey.
The high frequencies of J1 in Greece, Albania and Italy correspond to the high frequencies of J2, G2a, T and E1b1b, which also confirms a Neolithic to Bronze Age Levantine/Anatolian origin (probably not all in one time, but in several waves, some bringing more of one haplogroup than others depending of the exact place of origin; for example, the Etruscans seem to have had more J1, J2 and G2a, but less E1b1b and T, while the Phoenicians appear to have had particularly lots of J1, T and Q by Near-Eastern standards).[/QUOTE
From what I see in the map, a possible source of J1 in Albania at least, could be the Gypsy population. I notice that Hungarians have a hotspot, so do Romanians, Bulgarians that have a considerable gypsy presence. Since Gypsies originate in India they crossed arabian peninsola at some point.