The rules for interaction of replacement effects specify that
616.1. If two or more replacement and/or prevention effects are attempting to modify the way an event affects an object or player, the affected object’s controller (or its owner if it has no controller) or the affected player chooses one to apply, following the steps listed below.
616.1e Once the chosen effect has been applied, this process is repeated (taking into account only replacement or prevention effects that would now be applicable) until there are no more left to apply.
Both cards create replacement effects so when they affect you, you choose the order (also the steps 616.1a-616.1c are not relevant here). If you apply the Mask effect first, the Reflection effect has nothing to apply to - you exile one card. If you apply the Reflection effect first the Mask effect will replace both draws with exiles.
This is also mentioned in a ruling on Thought Reflection:
If two or more replacement effects would apply to a card-drawing event, the player who’s drawing the card chooses what order to apply them.
After digging into the rules I have to expand (and revise) my answer.
Basically there are no exceptions to either card. Thought Reflection causes you to draw two cards instead of one, whenever you draw a card. This is replacement effect #1.
Uba Mask replaces the action of any draw with the exiling of the card that would have been drawn. Replacement effect #2.
But: The moment both effects would become applicable, the affected player chooses, which one to apply first (616 in the rulings). If you choose to apply #1 first, then you replace the drawing of one card with drawing two cards. Then #2 kicks in and you exile both cards instead of drawing them. If you, on the other hand, choose to apply #2 first, you exile a card instead of drawing it. #1 then does not do anything, since you did not draw a card.
For more information on replacement effects, check the ruling.