I thought I would show you my versions of the flag from the two 'Swiss' regiments of British Foot which fought in North America (I was going to state "in Canada" but the de Meuron Regiment did join the advance to Plattsburgh in 1814, losing only 14 in that tepid campaign). They differ from the usual British colours.
So here is the de Meuron colours, showing the typical Swiss 'flames' in the odd lime- green, black and yellow combination in the regimental flag. The King's Color has the Union Jack with the motto on the St James cross. Note hidden from view canton on the regimental flag has the Union Jack - which when seen makes for quite the hideous display! Osprey MAA335 has a good color illustration of it.
and here are the De Watteville colours (ah, flags) for the unit which had quite the bad luck at the siege of Ft Erie in 1814, having substantial casualties in a magazine explosion during a British attack.