What the United States can learn from the UK:
Judicial independence is really emphasized in the Commonwealth courts, limiting the amount of political influence onto judges. This is the principle the CMJA promotes. Independence of the judiciary is extremely important in dispensing equal and fair justice for all. The U.S. practice of electing judges as well as political appointments, for the Supreme Court and State courts, doesn't allow for the same levels of independence that is required of the judiciary. The nomination and confirmation of Supreme Court justices is also politicized, with the justices having to be confirmed by politicians in the Senate.
The United States could also explore cost-shifting mechanisms like the "loser pays" system that is implemented into the UK system. Under this system, the losing party pays the costs of council for the winning party. This could limit the amount of trivial lawsuits brought about. Currently, each party in the U.S. pays their own legal fees.
What the UK can learn from the United States:
The UK could implement a greater use of juries especially in civil cases. They have largely moved away from jury trials in civil cases, while the U.S. places a greater emphasis on the use of juries, in both criminal and civil trials. Juries can act as a protection against a judge's bias, bringing community perspectives into the legal process. There is also currently legislation being debated to get rid of jury trial for minor criminal cases, which has been widely controversial.
The U.S. is also a little better at case management and the digitization of the judiciary, especially post-covid. Technology has been widely implemented into daily court functions, and while this is also true for the UK courts, the U.S. system has better administration of justice in this sense, with case information being transparent and available online for anyone to access.