It would only be down The Mall.
I'm not sure if you know London, but while nowhere could ever be entirely safe, it is among one of the easiest areas in the world to secure in terms of risk to dignitaries travelling down it. It's basically parkland and government owned and controlled buildings along nearly all of the route, and the few exceptions belong to organisations that fully cooperate with the security services.
There's a reason this route has been used for such parades for the last century.
It would be up to President Trump whether he wanted the parade in the carriage, and some previous Presidents haven't done so for security reasons. But it really is as close to a safe route as anywhere on Earth can be.
The route can also be lined with soldiers from the Foot Guards (and these men are now hardened combat veterans in many cases) with the Household Cavalry in attendance. And the Queen is in the carriage too - so they would be protecting her as well.
And it doesn't have to be an open carriage - the Australian State Coach and the Diamond Jubilee State Coach, despite looking just like the old style coaches are both of modern construction and while security details about them don't get published, it's pretty well understood they have the same type of protections as modern day VIP cars. These enclosed coaches have been used for processions on state visits in the past.
Your description sounds good but not good enough. Trump will not ride in an open vehicle down The Mall.