England, not Britain, and no-one would countenace a republic in 1605. It only happened OTL in reaction to years of Stuart absolutism and thanks to the helpful example of the Dutch. If nothing else it would break the union of the crowns, which everyone was massively politically invested in at the time.
Scotland just lost it's king as well remember, so referring to the problem as "British" rather than "English" is fine if one doesn't wish to ignore Scotland completely.