The Canadian's had their own counter plan:
https://en.wikipedia.org/wiki/Defence_Scheme_No._1 It would have been a gigantic failure. The plan was to try and mobilize quickly, seize Buffalo, Detroit and Seattle, and hold on for dear life until the BEF arrived. But the British were clear from the beginning that they were never going to send significant reinforcements to Canada: the ocean was too large, the USN too strong, and the Canadian plains too vast for the British army to be able to effectively defend. At most they might try and send some troops to defend Halifax as a key naval base. So the Canadian plan ensured that their best troops would be lost quickly, that the Americans would be super-pissed off and unified, all for an ending that would never come. So while it might have made sense militarily, it could never have worked politically.
The US war plan for the UK was similarly weird: according to Miller, _War Plan Orange_ War Plan Red was the result of a deal between the US Navy and Army. The Navy wanted War Plan Orange (war with Japan) and so they let the Army write War Plan Red (the UK). Which was why a war between the two mightiest naval powers on the planet in 1925 called for the US Navy to be on the defensive, at most seize the Bahamas, Jamaica, and Halifax to try and deny them as bases to the RN, and then the US Army would invade a separate nation (Crimson aka Canada) that would probably try and be neutral in the war!
Basically, from what I can tell, the US, Canada, and UK were putting their best war planners on the likely threats, and putting their less experienced and good planners on these war plans. Because War Plan Orange was, at least if you squint, how the US defeated Japan. And the Canadian and UK's war planners for mobilizing in World War Two and sending them to France did a bang-up job. It was just these plans that were not thought through and would have been disasters if implemented.