New Connexion Street itself is named after the New Connexion Chapel. An Ordnance Survey map from 1877 shows a shop on Trelowarren Street with the small courtyard and chapel (labelled New Connexion) behind it. The chapel vestry was at the back of the chapel.
On Boxing Day 1881 at a bazaar and social meeting held on the premises, the upper floor of the building collapsed, sending about 120 people into the void below. Fortunately no-one was killed.
This footprint remains on the map published in 1908 prior to the construction of the new United Methodist chapel which was built up to the front of that chapel. The Trelowarren Street Chapel Sunday School stands on the imprint of the old New Connexion chapel, and may even structurally contain parts of that original building.
There is evidence in the archives that there was continuity between the the New Connexion chapel and Trelowarren Street chapel likely with the Union of 1907.
You can read a bit more about the New Connexion society in Camborne on the Dictionary of Methodism in Britain and Ireland here.