I am an introducer to at least one bank in those jurisdictions and yes it is possible to open personal bank account remotely, provided that you can meet their minimum deposit criteria
Nothing strange about remote account opening in Luxembourg, Switzerland, Singapore, and Liechtenstein for personal accounts (assuming there's some degree of wealth and client has a low risk profile). UK is a little less clear, you might end up in Isle of Man, Jersey, or Gibraltar instead in many cases.
Introducer isn't strictly necessary, but can help get you past the initial filters.