The famous reconstructions of Qieyun which was a rhyme dictionary published in 601 AD was the product of not just one dialect, but sourced its data from dialects from a number of areas. Scholars today believe it to be a compromise between the literary pronunciations of northern and southern pronunciations of the NanbeiChao (Northern and Southern Dynasty) period.
This is why one can derive a common ancestry between all the Chinese dialects, such as Hakka, Yue, Min (in its various forms), Wu, Xiang, Gan and the Mandarin dialects. Qieyun was a compromise, a Middle Chinese record of many pronunciations centered around the northern and southern courts of Yexia and Jinling respectively.
Old Chinese which was derived from Middle Chinese would similarly be a mishmash, because of the extrapolation involved getting from a fixed recorded point in Qieyun. and the use of phonological rules to get to the OC reconstruction. There are many reconstructions of both MC and OC, the reason being, each scholar who derive them take a slightly different premise and arrive at slightly different pronunciations. Reconstructions like Karlgren, Pulleybank, Chao, Baxter, Li etc...
Hakka can derive a tradition from MC because it has regular phonological correspondences with MC. For instance, we know that there used to be voiced initials called QuanZhou initials. In Hakka they survive as aspirated initials. Hakka retains the majority of the endings represented in QY, that is, it has syllables which correspond to the rhymes ending in -m, -n, -ng, -p, -t, -k. Where it has characters which are no longer pronounced in the same tone category as those listed in Qieyun, there are regular phonological correspondences between the class of characters which have entered another tone.
All dialects of Chinese exhibit these things. There are regular rules for the way a tone splits depending on the voicing of an initial, and even the derivation of modern initials from QY which depend on the medial and rhyme.
Given the fact also that Hakka retains many features not seen since before the mid-Tang period when it is believed that such features changed, such as the appearance of the bilabial fricative inital f, and its ancestor p which in combination with MC medial w gives rises to modern pronunciations of f- in most dialects, Hakka retains words which are spoken with the bilabilal plosive p.
For instance, the reading is modern Hakka found between //, whilst M. stands for mandarin
fat /p'ui/ M. fei
fly /pui/ M. fei
give /pun/ M. fen
nightsoil /pun/ M. fen
to place /piong/ M. fang
float /piau/ M. fou
It retains a dental fricative although devoiced
middle /tung sim/ M. zhong xin
know /ti/ M. zhi
It also retains features such as vowel dipthongs which are also a feature of Mandarin, but less so in Cantonese, the latter exhibiting long vowels instead. In fact, Hakka has multiple readings, featuring colloquial and literary readings,
Spring gravesweeping /ts'iang miang/,/ts'in min/ M. qingming
star /siang/,/sin/ M. xing
Whatever Hakka ancestry was in the past, Hakka of today are part of the Chinese sphere of culture. It may exhibit one facet of the past, but it is not divorced from the rest of Chinese culture.
As for whether Korean or Japanese ancestry was involved, I would say not likely in the last couple of millenia. The general areas from where Hakka are said to come from (zhongyuan) or Central Plains, is well within the border of China. To the best of my knowledge, there has never been a mass migration of Japanese to Chinese shores until the last 100 years, and the scholars who came to China during the Tang would not be significant anyway. We also should bear in mind that Tang China was a cosmopolitan place, with people from all over the world, and during its time, it had many different faiths whilst not particularly espousing one.
Hakka are Chinese.