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T&W is Tudor & Whitford of Dublin. This mark was registered by Richard Whitford who became free of the Dublin Goldsmiths Company in 1802 and died in 1824.
If I'm reading these sub-notes correctly (under the Irish date chart), then the shield shape around the marks here is used 1797-1807, thereby narrowing the date range for the spoon.
From observation of Dublin marks the cut corner rectangle was actually in use from 1794 and the duty mark was introduced during the 1807/08 assay year, giving a slightly wider window. However T&W's working dates were, according to Bennett, the authority on Dublin silver, 1802-1824, so we can assume a date range of 1802-1807.