Both came during the Neolithic, with the spread of agriculture from the Balkans to Germany (LBK culture). Both haplogroups were confirmed in
ancient samples from the LBK culture in Germany, although both at lower frequency than today.
Haplogroup J has been associated with increased heat production by the mitochondria, therefore providing an evolutionary advantage for northern climates. This may explain why this haplogroup became so common in Scandinavia, and especially in Norway (12%). In comparison, Middle-Eastern Y-DNA (hg E, G, J) make up about 10% of the population in North Germany, 5% in Denmark and 2.5% in Sweden and Norway. MtDNA frequencies are much more likely to vary in time if they can confer an advantage in a particular climate.
Haplogroup T is also extremely common in the Eurasian steppe and indubitably has connections with the Indo-Europeans, as attested by the ancient DNA from the Bronze Age in the Pontic steppes and Central Asia.
Incidentally, haplogroups J and T account for about 20% of the lineages in Italy, so it isn't little. Keep in mind that Middle Eastern lineages also included many subclades of H and K as well as haplogroups that are rarer in Europe like N1a, X2, U1, U7, U8 and U9.
Note that mtDNA haplogroup T, N1a and X2 have a strong affinity with the Caucasus, and surely correlate with Y-haplogroup G2a (and to a lower extent J2). Haplogroups U1, U8 and K peak in Anatolia and northern Mesopotamia (homeland of J2), but are also found in the Caucasus and beyond.
Haplogroups T and K are both quite common in the Eurasian steppes and surely represent the maternal counterpart of the migration of R1b1b from Anatolia to the Pontic steppes. This would explain why both haplogroups are so common in Europe and Central Asia, despite having originated in the Middle East. In Europe, K reaches its highest frequency around the Alps, in South and West Germany and Belgium, i.e. areas where R1b1b2a first settled, and which evolved into the advanced Celtic cultures of Hallstatt and La Tène.