I don't think by "Going Swiss," the UK would become an exact copy of Switzerland policy wise.
The British Monarch would still be head of state of the dominions.
However on the issue of "going neutral" I believe the focus would be on the colonies, and not on joining in any grand crusades against communism led by America.
Britain would most likely still partition India, but with more preparation, the Kashmir Valley issue might be avoided before it becomes a problem.
Egypt and the Suez might not become an issue if Nasser can be kept out of power, Egypt is enticed to become a member of the Commonwealth(not likely), or Nasser does come to power, but is quickly crushed by fast acting British troops stationed near by.
Other colonies in africa would be granted independence, but that process would take longer because the process of granting them independence would be more controlled, not chaotic, and might lead to more stable countries late on.
On the issue of Britain as a tax haven, I don't think that is possible. The best possible outcome on that front is to keep the Tories in power as long as possible so as to prevent the modern welfare state from emerging.
So to reiterate, Britain going Swiss would not involve Britain following the swiss model 100%. The UK would focus on its empire and perhaps reform its empire into an earlier British Commonwealth.
Wishing you well, his majesty,
The Scandinavian Emperor