The creature can move at half speed to avoid both effects.

Neither ball bearings nor caltrops state that a creature's speed is physically penalized, just that a creature can choose to move at half speed to avoid their effects. Then we can handle each consideration separately.

First, ball bearings (emphasis mine):

As an action, you can spill these tiny metal balls from their pouch to cover a level, square area that is 10 feet on a side. A creature moving across the covered area must succeed on a DC 10 Dexterity saving throw or fall prone. A creature moving through the area at half speed doesn't need to make the save.

Is the creature moving through the area of ball bearings at half speed? Then they are unaffected.

Next, caltrops (emphasis mine):

As an action, you can spread a bag of caltrops to cover a square area that is 5 feet on a side. Any creature that enters the area must succeed on a DC 15 Dexterity saving throw or stop moving this turn and take 1 piercing damage. Taking this damage reduces the creature's walking speed by 10 feet until the creature regains at least 1 hit point. A creature moving through the area at half speed doesn't need to make the save.

Is the creature moving through the area of caltrops at half speed? Then they are unaffected.

An area covered in both ball bearings and caltrops is no different. Is the creature moving through the area at half speed? Then they are unaffected.

Note that if the creature chose to move greater than half their speed across the combined area then they would be forced to make both saving throws, one against the ball bearings and one against the caltrops, possibly suffering one effect or the other effect or both effects combined.

For simultaneous saves, there is an optional rule

Generally, when one is making a saving throw you are making them one at a time in the order they occur. That is the rule that saving throws operate under normally. However, if the saves somehow end up happening at the same time with no logical way to separate them, we do have an optional rule that will help.

In Xanathar's Guide to Everything there is an optional rule that a DM can use to adjudicate cases where multiple things happen at the same time:

Simultaneous Effects

If two or more things happen at the same time on a character or monster’s turn, the person at the game table — whether player or DM — who controls that creature decides the order in which those things happen. For example, if two effects occur at the end of a player character’s turn, the player decides which of the two effects happens first. (p. 77)

So, in this case, the player who is making the saves gets to choose the order of the saves they make assuming it is their turn. In the weirder case of the player moving through the caltrops/bearings on another person's turn, that person will decide the order.