How did they measure speed in the 19th century so they could issue the first speeding ticket?

In general, it is quite possible to gauge another's speed by following behind them in your own vehicle at a constant distance and take down the speed your own vehicle says it is traveling. This is how it was done before radar, and can still be done today. Where I live, last I saw it was being called the "pursuit" method.

Here's a sample ticket I found online. Notice at the bottom that there are boxes for the issuing officer to check to indicate whether the speed was clocked via radar or persuit.

Your assumption that there was a speed limit, measured im km/miles per hour, existed may not be true.

In a 1919 training book, based on the first national traffic laws of 1909, for policemen in Germany, the speed measurement was based on that of a horses gait.

Within a town:

• no galloping
• no trotting in heavy traffic areas

For Automobiles it is stated that the maximum is that of a trotting horse: 15 km or 300 to 320 paces per minute over a distance of 250 meters.

For heavy vehicles: 5km within a town, 7km outside.

It also states: The exact determination, even for a distance of 200 meters, is to be considered difficult because a watch is not very precise and the exact start/end points cannot be determined reliably and therefore a great amount of leeway must be given.

Since Motorcars only came into use in the United Kingdom in the early 1890's, most people in 1895 would probably gauge a speed based on a horses gait, since the leg movement of the horse could be easily noticed.

Therefore a certain speed would be associated with one of the 3 major gaits (a fourth being a canter, 16–27 km/h (10–17 mph)):

walking	trotting	galloping
7 km per hour (4.3 mph)	13 km per hour (8.1 mph)	40 to 48 km per hour (25 to 30 mph)

• Photos taken by Eadweard Muybridge, first published in 1887

A statement that the Motorcar was moving at the speed of a trotting horse could be accepted as a realistic estimate.

The police officer who arrested a president - The Washington Post

• ^{The Washington Evening Star in 1908 recounted the story of President Ulysses S. Grant's 1872 arrest.}

The police had been receiving complaints of speeding carriages. After a mother and child were run over and badly injured, Officer West was dispatched to investigate. As West spoke to witnesses, another group of speeding carriages headed toward him — including one driven by the president of the United States.
...
The president apologized, promised it wouldn’t happen again, and galloped away.

This implies that the Officer only looked towards the carriages and saw that they were speeding and charged accordingly.

Sources:

• Horse gait - Wikipedia
  • Eadweard Muybridge
• Die Polizeischule, Band I
  • Chapter 10: The important Traffic regulations
  • second edition 1919, Der Polizeschule Verlag, Dresden
• The police officer who arrested a president - The Washington Post

For this particular case, there is a (very) detailed account of the trial in the Maidstone Journal and Kentish Advertiser, 30 January 1896:

I.C. Heard ... was standing at the window of his cottage at Paddock Wood when he saw a horseless carriage go past with two persons in it. The carriage was proceeding at the rate of about eight miles an hour ... at once went in pursuit, and caught it up in the Maidstone-road about half an hour afterwards. It was proceeding at a fast rate.

No information is given about how "the rate of about eight miles an hour" was arrived at. Another report in the Kent & Sussex Courier notes that it was admitted by the defendant, so does not seem to have been particularly contentious.

(Incidentally, "I.C. Heard" was "of the K.C.C." - he was a council employee, not a policeman.)

In general, there were two approaches which could be used. It seems likely that this case was simply an estimation by the witness, given the speed is quoted at a particular moment and seen from his window.

The estimation approach is that ... often, they didn't measure. They just had someone say "it seemed to be going about so fast", and if the witness was considered reliable, that was good enough. Remember that while cars were new, there were already plenty of other things on the roads to use as benchmarks. In the case of the "locomotive" limit of 2mph, it would be easy to say something was clearly going faster than it without needing to exactly measure the speed.

Looking through the Old Bailey trial records - which don't seem to have plain speeding cases but do have various "injury by dangerous driving" etc - it seems that in many cases an estimate was taken as evidence, without requiring a measurement.

This approach had been in use before motor vehicles came in; for example, in an 1895 case, we see police constables giving evidence in a way which indicates a familiarity with estimating speeds:

it was going eight or nine miles an hour; I said so before the Magistrate—I might have said "Seven, more or less"—

I know this road well—eight miles an hour would not be a swift pace to drive a cart there at half-past ten at night—I should not stop a man at that pace

This 1907 case has several witnesses giving evidence about the speed, and while none of them were policemen, it looks like it was treated as something that an experienced observer could reasonably estimate. (Most were bus drivers or conductors; one was a passenger who simply noted that it was going faster than a bus ever does). The magistrate was quite prepared to accept the estimates from the witnesses and deliver a verdict citing the speed as a proven fact - "the evidence showed that this car was driven along Oxford Street, one of the most crowded thoroughfares in London, at at least 16 miles an hour".

In a 1911 case, one witness took a different basis for comparison and described a car as "going as fast as the train I generally go home in—between 25 and 30 miles an hour."

The alternative approach is measurement. This is referred to by the defendant in that last case, talking about their previous conviction:

J. H. DAY (recalled, further cross-examined). I was driving this same car when convicted at Sutton about August last year for exceeding the speed limit. I was timed for only one-eighth of a mile. It was alleged that I was travelling 26 miles an hour. It was not in the Sutton High Street; it was in a country road.

In other words, a policeman timed the driver over a set distance, and calculated the average speed from that. (It does not give an instantaneous speed as a modern radar gun would.)

It is possible that this could have been used in the Paddock Wood case, given that the witness knew how long between setting off in pursuit and catching the car, and presumably could work out where he did so. However, it seems unlikely as this was not mentioned in the reporting of the trial.

This approach was in use as early as 1902, judging by this mention in a Kipling short story - Steam Tactics, first published in December 1902.

He stopped, for full in the centre of the Linghurst Road stood a person in pepper-and-salt raiment (ready-made), with a brown telegraph envelope in his hands.

"Twenty-three and a half miles an hour," he began, weighing a small beam-engine of a Waterbury in one red paw. "From the top of the hill over our measured quarter-mile—twenty-three and a half."

In other words, he had timed the car from "the top of the hill" to a certain point, and was able to give a pretty precise average speed as a result. (The "Waterbury" was a cheap watch, and the beam-engine is just the author being rude about how primitive-looking it was. They proceed to take their extensive revenge on the policeman, who has forgotten to bring his badge...)