You'd have to stop the game, save it and reload it.
That works fine if I reload the game in single player when the game year is 1940 or later, but not in multiplayer. There will still be dirt roads if I save and reload the game in multiplayer after 1940. I found a way to force it to switch to asphalt when the game is loaded in multiplayer since I last posted, but the method can potentially break the game. I will list these steps at the bottom of this post.
I did this with a multiplayer game that had a starting date of 1900, whose current year was 1970 before I did this. Prior to this, the parameter mentioned earlier was set to the default of 1940, but it would never switch to asphalt in multiplayer. However, it switched to asphalt just fine in single player without having to do the steps below.
1. Close OpenTTD and enable the scenario editor if it isn't already enabled.
2. Start OpenTTD and load the save file in question in single player.
3. Open the NewGRF Settings window and change the value for the "Switch to asphalt in year:" parameter for the American Road Replacement Set to 0.
4. Let one in-game day pass, then save the game.
5. Quit the game and load the new save file in multiplayer mode. You should see asphalt roads now, and hopefully the game didn't break in any way.