Cadwallader = Welsh "battle leader". Cadwallader was a Welsh Christian King who died in 682.
Cadwallon = Celtic "war lord"
Cadwell = Celtic "war defence"

Cadwallon Lawhir = one of the many who have been thought to have been the legendry "King Arthur"

I am not sure but I think Cadwallader=Cadwallon

The cross of Cadwallader is the "Cross Pattee Fitche" which is also known as the Cross de Maltha Caldwell and adorns the shield of several English Caldwell families.
Someone must have thought that there was a connection (even just the similarity of the names?)

The great seal of James I of Scotland had the banners of Cadwallader (azure, a cross patte fitche or) and King Edgar (azure, a cross patonce between four martlets or)
What the significance is I don't know. Cadwallader was a North Welsh King and the Stewarts had orginally come from Shropshire to Scotland.

Open for debate.