They're still a Bladesinger
If you take a real persnickety reading of the rules, the SCAG says specifically that:
Only elves and half-elves can choose the bladesinger arcane tradition. In the world of Faerûn, elves closely guard the secrets of bladesinging.
In order to choose the bladesinger arcane tradition, one must be an elf or half-elf. But you only choose your arcane tradition once, upon reaching second level; thereafter, you just have it. A bladesinger who is reincarnated as a non-elf remains a bladesinger because they were an elf at the time they chose their tradition and being reincarnated does not require them to choose those options again.
This works perfectly well with the lore given; the justification for the restriction is that "elves" (well known monolithic entity that they are) simply don't teach the secrets of the tradition to non-elves, not that it's anything which is innately tied to elven physiology. To be a bladesinger is to know these secrets, and you don't forget them just because your biology changed.
The rules also don't make any presumption that a character needs a trainer in order to advance their class features as they level up, so the reincarnated bladesinger should be able to continue progressing in the class even if there are no elves who would be willing to teach them the secrets of bladesinging in their new form. In the fiction, I would judge that once initiated into the ways of the class, it is possible for the character to make further progress in their class abilities by practice, self-study, experimentation and discovery.