I am a halfling monk who has recently acquired boots of striding and springing. As a halfling I have a base movement of 25 feet; however, as a monk (currently level 4) with Unarmored Movement, I have an extra 10 feet of movement, plus an extra 10 feet from the Mobile feat (PHB, p. 168) - for a total (pre-boots) of 45 feet.

The boots of striding and springing raise your walking speed to 30 feet if it is less than 30 feet.

Does this speed increase to 30 feet take effect before or after the other modifications? In other words, would my total speed be 50 feet, or remain 45 feet (as is the case without the boots)?

The boots are applied to your speed, after other (permanent) additions

The boots give you speed, not movement. Speed is effectively the amount of movement you get per turn. Therefore the boots give you no benefit unless your speed is reduced by something else.

Because the movement speed from monk and from a feat are permanent and not optional or conditional in any way, your walking speed is 45 when you put on the boots so they do not boost it.