Thanks for the replies!
Yeah I think a gutter is probably the best solution, but concerned about how much a bespoke curved one would cost! All the timber is treated, however the door every year gets extremely stiff and we're now at the point where we need a new one...
A small canopy above the door could work but I wouldn't want it to be too overbearing as it would take away the nice curved proportions of the facade and roof. Would that need drainage/ a gutter as well?
Here's a picture of the whole house, located in Kent :)