For example, could a wealthy businessman travel from New York to London for matters of important business? Or did you in some way need to be connected to the war effort to be able to travel?
Neutral aircraft still flew the Atlantic. A typical path for a business man from a non-involved country (of which there were a few) would have been to fly into Spain and arrange transport from there.
(thinking specifically about London)
Or did you in some way need to be connected to the war effort to be able to travel?
Remember that the UK very quickly moved to a wartime economy. Almost every aspect of British industry was subsumed to the war effort. The government took more or less complete control of all transportation : roads, rails, ships and aircraft.
As /u/Noisy_Rhysling says neutral aircraft did cross the Atlantic. If you were heading to an Axis country you would be fine once on the continent. Getting into the UK from Spain however could be still be dangerous.