Using such as a historical source of accurate information is rather shaky.

I can however help with one of your questions:

"what war are the British soldiers trying to press Arthur MacBride in?"

The British Army have never "pressed" anybody and they have never had the legal power to impress anyone into service. The Royal Navy on the other hand did have such power. As to which war? IIRC the earliest mention of the song was around 1820s, 1830s somewhere down in Somerset or somewhere in the West Country so most likely candidate would be the Napoleonic War towards the end, 1814 or 1815.

A clue in the song contained in the following lines

For you would have no scruples to send us to France,
Where we could be shot without warning

Very few wars between Britain and France ever involved a British Army actually fighting on French soil.