Okay, this will be somwhat hard to pin down as the colonies had substantially different judicial system structurally, and of course legislation as such. Initially, one has to seperate between proprietary and royal colonies, but after this disctinction post-restoration, or rather slightly later in some case, post-glorious revolution ( for example, Maryland proprietary charter was annuled in 1692, and became a royal colony ), but the characterization to the amount of British interference would be hard to concretize, since interactions were typically of a more informative nature, and recommendations were not, one could say, "binding" and were thusly not properly implemented, if implemented as such. Certainly, Council of Trade and Plantations, with the Board of Trade, by the early eighteenth century, somewhat increased. Then in 1696 Navigation Act by the British parliament created (vice)admiralty courts in the colonies ( which were rather lukewarmly received, specially later ), and lastly, Committee for Hearing Appeals of the Privy Council, which received appeals qua appeals ( as complaints were also received at the Council of Trade and the Board of Trade, but of different kind, and not as parts of "judicial hierarchy", but rather systematic and legislative issues, unprofessionality ( qua little formal training ) of judges or lawyers initially, complaints of contrarity to Common law etc. ) The latter two exerted some pressure on codification and systematization, in some instances changes before enactement, of legislation, but again, colonies differed, so here a colony-specific approach is needed. Perhaps another important case can be mentioned, Privy Council in 1727, case Winthrop v. Lechmere, held that Connecticut intestacy statute was invalied insofar as it was contrary to the laws of England. But this case is an exception, and it was left unenforced. And a few years later, the Council did not annul Massachusetts intestacy statute, that was, on the merits, almost the same. Also, governors of the royal colonies had to submit legislative acts to the Council for approval, butit generally avoided confrontation, even when statutes plainly contrary to the English Common law were enacted. ( I am not familiar with any detailed study on this point though ).

If we go over the criminal matters first, the appeals were not possible, or only as far as the accused was sentenced by the general or superior court, the governor and his council would consider appeals on the basis of writ of error, not facts of the case.

For other matters, Privy Council (Committee for Hearing Appeals) was the last appellate organ, and this database made by Ames foundation with Harvard law compiled existant documentation, and for the most thorough study, see Joseph H. Smith’s Appeals to the Privy Council from the American Plantations.

( And perhaps to address a point, the reception of Common Law varied, but early colonies certainly, or even in the early eighteenth century, substantially differed if we take the comparison with English Laws, as the colonies had their own legislative bodies. Except few exception, where the Parliament enacted legislation which was applicable in the colonies ( for example, already mentioned admiralty courts ), surprisingly little, although influence as such, in the broad strokes, was there, of course, simply by the origin.

It might be much easier to address specifics though, but I´d gladly say more on any issue raised above.