I confess that what I know relates to the Wars of the Roses and not the HYW, which while very close in time, are quite different propositions as far as liveries go. From what I can gather, the need to display 'national signs' clearly overrode other considerations, but most leaders were also vain types who wanted to display their power too.
Early WotR sources (and the article Dez offered) mention 'bends', which were a shoulder sash and magnates like Buckingham ordered these in the thousands. These were presumably worn over 'national livery' or armour/jacks displaying the St. George's cross, as secondary identification. The other method would be to supply coloured hoods to your men; Archers from Cheshire and Flintshire wore green and white c.1400. Having coloured hoods over the white 'national' livery, or bare jacks/akhetons etc, would certainly do the job to my mind.
I would probably opt for any of the common colours of the medieval era for liveries and whatever you opt for is likely to be 'historical' for someone. 'Madder-Red', 'Woad-Blue', 'Wool-White' are cheap to produce, with greens and subdued yellows bringing up the rear price-wise. Black also features (thanks to black sheep). If you had cash to burn then stronger hues of these colours are options, as does a sort of a bluey-purple and a crimson (as opposed to an 'Imperial Purple), produced by over-dying.
Black-Red, Red, Red-White, Red-Blue, Blue, Blue-White etc. are all common combinations in the WotR and I don't see why they would be any less common earlier; barring some attached meaning to the colours that we are not aware of. Some colours were associated with saints and in France blue was the colour of St. Denis and red was the colour of St. Michael (iirc), so these appear a lot in illustrations and went very well with the white cross of France.