Voltage at a point with OR-combination of diode
You were curious about the behavior and I think it's been answered in the comments, already. But you also asked about the case where one battery voltage (or the other) is applied first, and how the circuit might respond then when the other voltage is applied.

To help clarify all this, take a look at the image below. In it, I've arranged to have \$BAT_1\$ rise slowly up to its \$10\:\textrm{V}\$ value, while \$BAT_2\$ remains at \$0\:\textrm{V}\$. Then, once \$BAT_1\$ has risen and held it's value for a time, I allowed \$BAT_2\$ to rise slowly up to its \$20\:\textrm{V}\$ value, still keeping \$BAT_1\$ at its active value. Then I reversed the process so that \$BAT_2\$ rose upward first and \$BAT_1\$ only later was allowed to rise upward.



The BLUE curve is \$BAT_1\$ and the RED curve is \$BAT_2\$.

The GREEN curve is the voltage across the resistor. In the first period you can see that it goes to \$10\:\textrm{V}\$ and then goes to \$20\:\textrm{V}\$, as I said it would. In the second period you can see it goes straight to \$20\:\textrm{V}\$ and stays there, ignoring changes on \$BAT_1\$ that occur within that period.