Why does the Vimy have neither military serial or civil registration ?

Because the first permanent British civil register came into force on 22 July 1919, a month after the Trans-Atlantic flight. It was a civil aeroplane at the time, but the requirement to have a registration (or a certificate of airworthiness) wasn’t there yet. A temporary scheme had been started on 1 May 1919, but as far as I can tell the Vimy wasn’t registered in the K-sequence that was used. And after its undignified arrival in Ireland, the Vimy would not fly again so there was no need to retroactively register it.
More about this here: https://en.wikipedia.org/wiki/United…t_registration