The independence itself is not less lemongrove , but the psychological independence is less. They are too close to the action. Until 2009 the House of Lords fulfilled the role of supreme court. and the Lords of Appeal automatically were in the House of Lords. 2009 is only 10 years ago.
All the main courts are in London and, for barristers, who until recently provided all our judges, the Inns of Court where they train are in London, as are many barristers.
When power in this country is so often accused of being London centric I can see that the London-based English judiciary, given their reputation for independence would be much more wary about being pulled into national governance disputes. The English judges decided that the case was not justiciable (capable of challenge) in the courts because to seek the prorogation of Parliament is political matter. They then granted a leave to appeal to the Supreme Court, which has yet to rule on the subject.
I think the way it is working out is the best way - that the challenge comes from the Scottish Courts not the English Courts.