How long does it take, in game, for a party of characters to go from level 10-20?

Depends entirely on the campaign:

If leveling is based on Milestones, then it is directly tied to the length of the campaign for which those levels are appropriate.

If leveling is based on awarded experience, then a more combat heavy campaign could result in very fast leveling, while a more social campaign might result in very slow leveling.

For games played in Adventurers League style, I believe characters level up every 3 sessions.

Leveling is kind of like burning a candle: the brighter the flame, the quicker it burns.

Pace yourself according to your priorities.

It also depends on how much downtime you do.

And also which level you start on. 

There are no "spending downtime to grind experience to level up without an adventure" rules, so the answer is that characters might never reach level 20 in their entire lifetime. It's entirely possible (more likely even) that they die or retire long before they reach level 20.

It varies with the dm. But in the current season of Adventure League it would be 10 days. Since you can level up after each module. In homebrew this is more of question to your specific players.