Either use the simplified PHB rules or do the real math

There isn't necessarily a 'right' way here, but you do have to account for the height as part of their distance because that is how far the target is from the archer.

The quick and dirty

This is done by utilizing the Grid Variant option found in the PHB on page 192:

To determine the range on a grid between two things—whether creatures or objects— start counting squares from a square adjacent to one of them and stop counting in
the space of the other one. Count by the shortest route.

This doesn't make any real world sense, but in the base system, there is no difference between a target 110' directly below and one 30' away as well. They're both considered 110' away.

But it's easy. If you're looking for true distance, then you've got some math to do.

Pythagorean Theorem to find true distance

Or you can call Pythagoras and use his theorem (A²+ B² = C²) to get a more exact value. In this case A would be the mast height, B the horizontal distance from base of mast, and C is the hypotenuse for true distance.

The true distance would be 112 feet based on the hypotenuse length.
The Theorem system matches that optional grid variant rules in the PHB (pp 192):

DMG Offers an other Variant:Diagonals option (pp252)

When measuring range or moving diagonally on a grid, the first diagonal square counts as 5 feet, but the second diagonal square counts as 10 feet. This pattern of 5 feet and then 10 feet continues whenever you’re counting diagonally, even if you move horizontally or vertically between different bits of diagonal movement. For example, a character might move one square diagonally (5 feet), then three squares straight (15 feet), and then another square diagonally (10 feet) for a total movement of 30 feet.

Carcer actually has simplified how this would work and is a great suggestion for getting something more realistic and getting it quickly:

The distance according to the variant diagonals rule can be very quickly calculated by taking the longer of the horizontal or vertical distance and adding half of the shorter distance, rounding down to the nearest 5ft multiple - e.g. a target 110 ft. down and 25 ft. away (or 110ft. away and 25ft. down) is 110 + 12.5 ~= 120 ft. distant.

Be aware of other issues - and be consistent with whatever you choose

The DM may also want to consider the use of partial cover as the archer has to shoot through the rigging around the mast - but that's up to the DM.

Motion of the ocean may be another issue to consider. The potential rolling seas as well as the crows having a much stronger motion feeling due to it's height may impose disadvantage on the shot should the DM think it was worth doing so.

Also be aware of any darkvision distance issues if fighting at night.

Whichever method you choose, I do recommend that you use the same methodology in all cases whether PC or NPC. Make the rule known and agreed upon so that the players understand the implications.

Even though it's extremely common to do so, 5th edition assumes you are not playing on a grid, by default. Using a grid is proposed on PHB 192 as a variant rule. In that rule, it says that moving to any adjacent square (including diagonals) counts as 5 feet, period:

Entering a Square. To enter a square, you must have at least 1 square of movement left, even if the square is diagonally adjacent to the square you're in. (The rule for diagonal movement sacrifices realism for the sake of smooth play. The Dungeon Master's Guide provides guidance on using a more realistic approach.)

This means, regardless of actual math, that we can measure your distance traveled as two straight lines, distance forward, and to the side. Whichever of this is longest will be the equivalent of how far you've moved.

You would measure other distances the same way, like how far away a flying creature is, or how far we need to shoot downward at a target.

A simple example of this is if I move forward 10 feet and to the side 5 ft (2 squares by 1 square), I've moved 10 feet.

So for your example, if he is 110 feet up, but only 25 feet away horizontally, the distance would be measured as 110 feet.

This is used for simplicity's sake but is not mathematically accurate. Aware of this, the designers proposed an alternate method in DMG 252 (as mentioned above):

Optional Rule: Diagonals

The Player's Handbook presents a simple method for counting movement and measuring range on a grid: count every square as 5 feet, even if you’re moving diagonally. Though this is fast in play, it breaks the laws of geometry and is inaccurate over long distances. This optional rule provides more realism, but it requires more effort during combat.

When measuring range or moving diagonally on a grid, the first diagonal square counts as 5 feet, but the second diagonal square counts as 10 feet. This pattern of 5 feet and then 10 feet continues whenever you're counting diagonally, even if you move horizontally or vertically between different bits of diagonal movement. For example, a character might move one square diagonally (5 feet), then three squares straight (15 feet), and then another square diagonally (10 feet) for a total movement of 30 feet.

If you aren't using the Grid variant rule, you would just measure the distance from point A to B.

You can also, of course, actually do the math as NautArch suggested in his answer. It comes down to what you and your players can agree on. But either way, whatever rules you set for measuring diagonals should be used for all cases of measuring diagonals. As long as you're consistent within your own game, that's what matters.

There is no solid right way to do this, but here are three ways this can be adjudicated.

Pythagoras

Using the Pythagoran theorem of $a^2+b^2=c^2$ you would obtain a distance of 112.8 ft. This is the most accurate method, but is going to necessitate calculations, and your mileage may vary but I prefer not to large squares and roots in my head and having to resort to a calculator all the time might slow your game down.

Simple sum

If you just add the distances together (giving 135 ft) you can very quickly estimate an upper bound for the distance. This can be very useful because if that estimate puts it in range then it is going to be and more accurate calculations need not be performed.

Adapt the Optional Diagonal Rules for playing on a grid

In the optional rules¹ presented on page 252 of the Dungeon Master's Guide every other diagonal move in a 5 ft by 5 ft grid costs 5 extra feet of movement where a normal diagonal costs 5 feet. You could then move diagonals counting 5 feet per and add an extra 5 for every other moved. However to make it simpler we can observe that a move 10 ft vertically and 10 ft horizontally would cost 15 feet when moved diagonally. A resultant rule to estimate a angular distance is that for every 10 ft moved both diagonally and vertically, 5 feet is subtracted from the summed distance.

With our specific example the vertical distance is 110 ft and the horizontal is 25 ft. The summed distance is 135 ft, but there is 2 '10-feet' moved by both so 2$times$5 feet is subtracted. The estimated distance is then 125 ft. (The number of '10-feet's would be equal to the smallest of the distances divided by 10 and rounded down.)

This is between the two other estimates and has the advantage of being fairly quick to calculate (much more mental-arithmetic friendly than Pythagoras) and consistent with other movement (if you're playing on a grid).

¹: It is an optional rule for the variant rules presented on the sidebar on page 192 of the Player's Handbook.

The shorthand for determining distance is in the PHB 192, "Variant: Playing on a Grid":

To determine range on a grid between two things - whether creatures or
objects - start counting squares from a square adjacent to one of them
and stop counting the the space of the other one. Count by the
shortest route.

Because of the 3-dimensional nature of being above them, if you wanted to do the Variant RAW then it would most likely be be the squares of space down the mast (22 squares for 110ft), and then up to 5 squares for the remaining 25', since the first 5' of that would be the adjacent square you begin counting from, making the range anywhere from 115' to 135' in the basic rules.

Personally, I'd recommend going for NautArch's Pythagorean approach and incur half-cover to both the gunslinger and the targets for realism and to not have your player have a "fish-in-a-barrel" scenario of free shots to anybody not coming to a sea fight with bows.

This GM always uses the following houserule:

Vertical distance doesn't affect the higher shooter, whilst it is added to the horizontal distance for the lower shooter

Thus your character in the crows nest will be able to shoot at the creatures as if they were 25' away, whilst those on the ship shooting at him have to do so as if they were 135' away.

This gives the higher combatant an advantage, as shooting up is always harder than shooting down, whilst it avoids any complexity from trigonometry or the diagonal rule.