The river whose source is traditionally marked at Thames Head in Gloucestershire is the River Thames.
Thames Head is a group of seasonal springs near Kemble and Coates in the Cotswolds. The Ordnance Survey and the Environment Agency traditionally identify it as the source of the Thames, and a marker stone at Trewsbury Mead commemorates the headwaters. The springs are fed by limestone aquifers, so their visible flow can vary with groundwater levels.
The Thames then crosses southern England through places including Oxford, Reading, Windsor and London before entering the North Sea through its estuary. At about 215 miles, it is the longest river entirely in England, although the River Severn is longer in the United Kingdom as a whole.
There is a genuine source debate: Seven Springs, the source of the River Churn, lies farther from the Thames’s mouth and would produce a longer measured system. However, the question specifically says “traditionally marked at Thames Head,” making Thames the canonical answer rather than the Churn or another nearby stream.