I got curious and looked around. I looked in the Boston Globe archive. Most early mentions of pneumatic tires are in regard to bicycles. The first mention I could find of wagons was this, on Sept. 7, 1895:

Use For Pneumatic Tires

Wagon Thus Equipped Believed to be Employed by Burglars

The Cambridge police are using every means in their power to protect the citizens from the gang of crooks now infesting this city and its vicinity. Extra precautions are being taken, and several more policemen have been put on duty and patrol the sections most likely to be visited by thieves. They are dressed in civilian clothes.

[...]

The police are looking for a wagon equipped with pneumatic tires, which was seen in Cambridge a few nights ago, and in which were four men. A policeman noticed that the vehicle made but little noise as it was drawn along the street, and upon attempting to investigate, the driver whipped up his horse and got away.

A burglars' team equipped with pneumatic tires is something new, and affords great protection against detection, and against pursuit after being discovered.

An 1897 article mentions a carriage for Pres. McKinley made with solid rubber tires.

A Nov. 2, 1897 article mentions a proposal to create a law to put pneumatic tires on the wagons of coal teamsters working at night, as a way to discourage them from working into the night at all.

The next mention I noticed was from Jul. 22, 1898:

With Every Tender Care

Extreme Measures to Prevent Annoyance to New York's Popular Society Woman

Still seriously ill from the accident that befell her while riding to hounds, Mrs. W. C. Whitney is on her way to Bar Harbor, Me.

[...]

From her home on the Wheatley hills, near Roslyn, LI, she was taken to the railway station in a wagon, the wheels of which were fitted with pneumatic tires. [...]

To Mrs Whitney's eyes the wagon was not extraordinary, and she commented with a joyful smile on her ability to endure the shock of it.

A Nov. 23, 1900 article:

Horse Show's Day

Biggest Attendance and Best Exhibits

[...]

[...] The special class for fleet roadsters had 21 entries, and no less than 17 of them took part in the contest of the trophy offered by Dr H. H. Kane, president of the Road Drivers association of New York. These nimble trotters, all hitched to pneumatic-tired wagons with wire sokes and thoroughly equipped for the road...

My conclusion from this is that people in the 19th century were accustomed to tire-less wagons and generally saw tires on wagons as an extravagance.

When invading Russia in the 1940s the German army developed many wagons with pneumatic tires and ball bearings. Most of the German supply line was horse drawn at this time. The modern wagons were not designed for muddy Russian paths and many times the German soldiers took wooden wagons from Russian villages and used them instead.

Pnumatic tires are expensive, and have to be replaced regularly, especially back then. Flat tires were a much more regular occurrence back before the invention of modern steel-belted radials in the 1940s.

Also, a big part of the improved ride from the use of pneumatic tires is most appreciated at speeds higher than is practical for most horse-drawn vehicles. Realize that few horse-drawn vehicles ever travelled over 15 miles per hour -- the power source and condition of the roads wouldn't permit it.

So I sort of answered my own question. I found an issue of a trade journal: The Hub, Vol. XXX5, No. 1, New York, March 1, 1894

Pneumatic-tired horse drawn vehicles did exist in the 1890s. By 1892, it looks like pneumatics had come to dominate sulky racing, according to record books. New sulkies were being built using pneumatic tires, and many owners were retrofitting pneumatic tires onto their existing sulkies.

P. 341 features an illustrated advertisement for a pneumatic sulky.

P. 345: "The Hume Carriage Company, of Amesbury, Mass., are fitting over several sulkies for New York parties, fitting them with pneumatic tires.

P. 431: "…by early fall the works will be running full blast, turning out bicycles and pneumatic-tired sulky wheels…"

Following the success with sulkies (which are a very light vehicle), attempts were made to adapt pneumatic tires to heavier wagons and carriages:

P. 164 features a short article called "Pneumatic tires for carriages," describing how pneumatic tires may be adapted or retrofitted to larger vehicles. The article also acknowledges how pneumatic tires had essentially taken over the market in sulkies at that point):

The advantages obtained by the use of pneumatic tires on sulkies have naturally been followed by efforts to adapt the "wind cushion" to four wheeled vehicles.[…]The problem was to so fasten the tire upon the wheels as to withstand the side strains of city pavements and car tracks, and also to overcome the liability to puncture as far as possible, and to provide quick and easy means for repairs when puncture should unavoidably occur.

These issues are still familiar to modern-day bicyclists who use "tubular" tires rather than "clinchers," which were new tech in 1893:

The wheels are made as usual, with the single exception of a very narrow rim, over which is placed the steel tire which holds the rubber. This steel tire is of peculiar shape, coming down the side of the rim, and then, by an upward turn, forming a grove around the wheel. The air is inclosed in an endless, sausage-shaped envelope of thin elastic rubber; over thi is placed a thick, tough rubber strip, which comes down into the groves formed by the upward turn of the tire, and is fastened there by bands of wire on each side…

The existing wheels can, in fact, be adapted for use with pneumatic tires by cutting the spokes and replacing the rim:

No change whatever is necessary in the construction of the carriage, except the wear-iron, which should be spoon-shaped, or, preferably, made with a roller. An old set of wheels may be altered by cutting them down three inches and placing new rims upon them with the pneumatic tire.

P. 25. "Electric light in carriages," details how the use of pneumatic tires in larger horse-drawn vehicles is rare in France due to the cost of tires, while economic conditions in the UK are more favorable.

The omnibus and cab companies of Paris do not appear to exert themselves, or better said, use their purses in providing electric light or pneumatic tires for the benefit of passengers patronizing their coaches, as the more sharp-eyed London companies are now doing.

At Glasgow (Scotland) I read that a model omnibus has been made, which has electric light and the pneumatic tires. I can safely say that there are not more than a score of vehicles in Paris having pneumatic tires. These are owned by private persons only. Perhaps their costliness is the reason for their scarcity here. In london a pair of tires cost from £7 to £10, which means £12 at least in Paris, or say 300 francs.

P. 488 describes a carriage built by the owner of the Bidwell cycle company. He adapted the carriage to use his company's bicycle wheels.

He has pretty thoroughly convinced the New York City carriage manufacturers, by actual use, of the practicability if not superiority of pneumatic tires, and has faith in his ability to demonstrate to the entire carriage trade that a vehicle with the Bidwell tire will run easier and with less perceptible motion than any other, and with less than one-half the wear.

Should the application of pneumatic tires to carriage meet with the same success that it has on bicycles, and it now looks as though it might, the time is not far distant when the rattle of wheels over the paved streets of our cities will be unheard, and, but for the sound of steel-shod hoofs, our avenues will be as free from noise in warm weather as when covered with sleighs in winter.

And indeed it seems like the technology world was gripped by a pneumatic tire mania in the early 1890s, with people trying to shoehorn the technology into places it wasn't practical (as was the case with all sort of technologies over time):

P. 911 refers to a "pneumatic road skate." Which sounds like the forerunner of modern in-line skates (having one wheel at the heel and one at the toe). But using pneumatic tires. Modern in-line skates, of course, have solid wheels.

(I'm sort of reminded how a current issue of a magazine like Wired or Fast Company would stereotypically feature umpteen mentions of "social networking" or "3d printing" being used in contexts that will probably seem laughable a few decades down the road.)