Has to go to Chancery. The colonies are established by Letters Patent so any dispute between the colonies is actually only resolvable by reference to the Chancery in the first instance (to the mid 19th century) because it relates to actions of the Crown in establishing the colony in the first place noone else can make a binding decision.
All this nattering about the details of the British legal system completely misses the point. It
doesn't matter whether it's the Crown, the Parliament, the courts, Chancery, or the Devil Himself who is trying to run the affairs of a dozen and a half or so growing, economically active, and frequently interacting colonies a few weeks away by ship. It really does not matter one whit, not the slightest bit, who or what is trying to centralize everything in London. Just the attempt to do without any intermediate levels of administration between the colonies and the central government is going to lead to an increasing burden on the responsible organs to resolve
American problems and address
American affairs instead of spending time on
British affairs and
British interests, including those of the Empire as a whole.
This is simply not a state of affairs that can be sustained. London is going to
want the colonies to come together and federalize into common administrations that can handle inter-colonial affairs without always needing to have London involved. This will include courts, of course, but there are many issues which are
not problems for the courts or chancery which will almost inevitably become points of dispute between the different colonies. As I pointed out earlier, in the case of the Australian and Canadian confederations, significant issues included tariff levels (which require inter-colonial coordination because of the large amount of trade between the different colonies), common investment in and standardization of infrastructure such as long-distance railroads (in Australia this was connected to the gauge problem, in Canada to the construction of an all-Canadian transcontinental railroad), immigration policy (which, much like tariffs, needs to be coordinated due to the large amount of travel between different colonies), and similar
legislative areas of interest. All of these are important. None of them are, in general, for courts of any type.
The fact that these will nominally be created by Parliament (or, to use the polite fiction you mention, the Crown) is...completely irrelevant. So was the Confederation of Canada. So was the Federation of Australia. But in practice these countries almost immediately became independent actors that only referenced a limited amount of policy to London, and instead reserved most for Ottawa or Canberra. That was, after all,
the point.