Shortly after inventing his double flageolet in 1806, Bainbridge moved from Little Queen Street to 35 Holborn Hill which was to become his main residence (and probably workshop) until his death in 1831. It was then inherited by his wife who passed it on to Henry Hastrick who, in turn, worked there from 1835 until 1855. Holborn Hill is located on the outskirts of the City of London, and is so called because it runs from the valley of the (culverted) Fleet River, up towards the higher level on which Holborn and High Holborn are built.
The street of Holborn Hill no longer exists. In 1863 it was demolished to facilitate the building of the Holborn Viaduct which now links Holborn with Newgate Street on the same level. Holborn Viaduct does, however, follow the same path of the old Holborn Hill. It seems that most Georgian buildings on the site were removed at this time, including 35 Holborn Hill.
To get a sense of where the workshop was, we need to speculate, slightly, from the available information. We know from Bainbridge's 1810 patent case against Wigley that Bainbridge lived in the parish of St. Andrew Holborn. The parish church still stands and is at the West end of the street. If his workshop was at the other end, he would have been in the parish of St. Sepulchre-without-Newgate.
Towards the Western end of where the street ran, lies Shoe Lane, which runs behind the church. It is still mostly–Georgian gives a little bit of the flavour of what Holborn Hill might have been like (albeit on the level rather than on a hill). As for the exact location of the shop, this will require a more investigation at some stage in the future, although to give some perspective of what this area of London is now like, the main possible sites are the City Temple and its bookshop or the British headquaters of the international law firm Lovells.