Don't know whether Newcastle ever became not a new castle . Newmarket is still a new market in the context of when it became a royally granted market, hundreds of year ago. But there are old towns and villages called Newton and, I think, Newtown. A strange example of this is the 'new bridge', the Pont Neuf, in Paris. Naturally, it is the oldest bridge in the city.
More seriously, Harlow New Town incorporated what became known as Old Harlow, but appears to have become simply Harlow in 1955 when there was a change in local government whereby the old parish of Harlow became an urban district and that included the new and old parts.
Thanks for all your answers, some of which didn't help! I think FredPuli43 is wrong because I moved to Harlow in 1960 and it was called Harlow New Town for some years after that.