Setting the initial To-address in .../directions/ by passing a parameter via URL

Sure, it's easy, just add ?to=[your destination] to the end of the URL.

Example: https://www.openstreetmap.org/directions?to=London

Note:

• The destination string is interpreted (by Nominatum I think) so what appears in the "To" box might not be an exact match. In this case "London" becomes "London, Greater London, England, SW1A 2DX, United Kingdom"
• You can add more detail to the destination but you may have to URL-encode some characters, eg https://www.openstreetmap.org/directions?to=London%2C%20Ontario becomes "London, Ontario" which is interpreted as "London, Southwestern Ontario, Ontario, N6A 1G4, Canada."
• You can also use coordinates as the destination -- but OSM will still attempt to guess an address, which sometimes succeeds and sometimes doesn't, eg https://www.openstreetmap.org/directions?to=51.4918%2C-0.1588
• If the user has visited openstreetmap.org before, the map will not automatically center on the destination. It will show wherever the user last was looking at OSM. If you want to control the map center and zoom, you can do so with additional URL parameters like this: https://www.openstreetmap.org/directions?to=London#map=10/51.4818/-0.1414 (here, 10 is the zoom level, 51.4818 is the latitude, and -0.1414 is the longitude.)