So say I drop both Caltrops and Ball Bearings onto the same area. If a creature, for example an orc, moves across the area and doesn't wish to make any saves, does it move at half or quarter speed?

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.

No, moving at "half speed" is enough to avoid both Dex saves

The description of ball bearings states:

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.

And the description of caltrops says:

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.

If a creature moving through the area moves at half speed, then it meets the (identical) condition for both items, and avoids the need to make a Dex save.

If the creature doesn't move at half speed, and enters a square filled with both ball bearings and caltrops, the creature is forced to make 2 Dex saves: a DC 10 Dex save to avoid falling prone, and a DC 15 Dex save to avoid taking 1 piercing damage and being prevented from moving further this turn (and having its walking speed reduced).