We have known about "continental drift" and plate tectonics for a while before the advent of precise GPS. Continental drift theory was developed in the early 1900's (1920's I think). Plate tectonics theory was developed in the 60's.

Geologists in the early 1900's (and earlier) observed related fossil assemblages and rock groups on the margins of different continents separated by large oceans. Continental drift theory proposed that the continents were once contiguous. Measuring the distance across the ocean basins provided a distance of drift, but not a rate.

Scientists in the 1960's used magnetometers to survey the ocean floor (the magnetometers were retired sub-hunters from WWII). They observed parallel bands of seafloor with the same magnetic orientation and intensity. They noticed that the bands were symmetric on either side of large ridges in the oceans. Plate tectonics proposes that the continents are going along for the ride as oceanic crust grows and spreads from ridges. The scientists used radiometric dating to calibrate the magnetic bands with a magnetic reversal time scale.

We now have the distance that the continents are from each other, and ages for the bands of oceanic crust between them, so we can calculate a rate. For example, the oldest crust in the Atlantic is about 180 million years old, and it is found off the eastern margin of North America and the Western margin of Africa (~6000 km).

6000km/180 million years = ~3.3 cm/year (Averaged over 180 million years, this is a very rough calculation).

This is how fast Africa and North America have cruised apart on average over the last 180 million years.

I hope that this explanation is coherent and helpful.

Before GPS and real time measurement of rates of deformation via satellite, we relied on older methods of geodesy. One method was laser ranging: setting up two locations and measuring the time it takes light to travel the distance. Repeating that measurement gives the rate.

For longer time scale, we can use the magnetic stripes in oceanic crust that gets magnetized when it forms along an ocean ridge. So we get a record of + and - polarity which can be calibrated by dating rocks on land and correlating oceanic magnetic record with the record on land from a pile of volcanic rocks.

First off, continental drift is sufficiently slow that we can't really measure it even now, except by measuring the individual earthquakes that cause it. Inter-continental distances however could be measured very precisely with the advent of time-syncrhonous communication (first celestial measurements of time of day, telegraph, later radio). By measuring the rising, setting, and positions in the sky of celestial objects, you can tell exactly where you are on the Earth's surface at any point in time. Thus, by measuring the locations on the Earth's surface of, say, New York, and comparing that to, say, London, you could tell, very closely, the distance between New York and London on the globe.