Adding to the previous answers regarding the use of the hand to assess body temperature, there have been several works which are quite suited to answer this question, which I hope to summarise here.
In essence, the answer to this question is in knowing when were thermometers first used to measure body temperature. As it is with this information that we can then work out what methods were being used to perform this task PRIOR to the first use of thermometers in this way.
However, upon embarking on this enquiry, one would find that knowing what exactly is deemed a 'thermometer', or for that matter if it was used to measure body temperature, i.e. in the clinical or medical setting, is not exactly straightforward. Consider the development of thermometers below.
A. Time points in the development of thermometers:[1-4]
240-200 BC: Philo of Byzantium (and later Heron of Alexandria) describes expansion of air by heat
The advent of baro-thermometers or air thermometers: these are not thermometers per se, as both temperature and atmospheric pressure contribute to the expansion or contraction of the substance being measured (called the thermometric substance). There are four likely candidates for being the first inventor of this type of thermometer.
*The Galilean or Italian tube: in this version of the air thermometer, there is a single hollow bulb attached to a hollow tube and the tube and bulb are placed (with the bulb pointing up) on a flask (that is used to assess the temperature of a thing) filled with a thermometric substance.
** anywhere from 1592-1603: Galileo Galilei
** 1611- Santorio (or Sanctorius) of Justipolitanus
** 1617-1626: Robert Fludd
*the Dutch tube: in this version, the tube is attached to two bulbs, with one bulb, that is used to assess temperature, being placed lower than the other in a u-shaped arrangement. The tube is filled with water via a hole in the higher bulb.
** anywhere from 1598-1626: Cornelius Drebbel
*1620: Biancani of Bologna coins the term thermoscope
*1624: Jean Leurechon coins the term thermometer
the advent of liquid thermometers: these are closed tubes with the thermometric substance sealed inside
*not later than 1641: Grand Duke of Tuscany, Ferdinand II de Medici. He designs a (closed) tube filled with the thermometric substance and glass balls of different densities floating in the substance.
*1654: Accademia del Cimento (Academy of Experiments). They design a tube with the only the thermometric substance sealed inside.
Development of the liquid thermometer.
*1683: Robert Boyle proposes the need of standardised scales to be used in thermometers
*1724: Daniel Fahrenheit's scale. He fixed three main points of the scale to the temperatures at these phenomena: the lowest (0 degrees) being the temperature of a mixture of ice, water and sea salt, the second indicating a temperature of the mixture without salt, and the third being the normal body temperature (96 degrees)
*1741: Anders Celcius' scale on mercury based thermometer. He fixed his scale to two main points: melting ice (100 degrees) and boiling water (0 degrees)
*1743: Celcius's scale is reversed by Carolus Linnaeus, Jean-Pierre Christin and Martin Stroemer, independently. This is the basis of the current liquid thermometers in use.
*1866: Sir Thomas Allbutt creates a small compact version of the liquid thermometer
1868: Carl Wunderlich publishes his work describing the course of body temperature for different diseases, which allows clinicians to know the status or progress of a human body based (among other clinical factors) on body temperature, hence popularising the use of the thermometer.
B. Considering the above, how was body temperature assessed before the thermometer?
If, by this, you refer to the modern and current liquid thermometer, it would suggest that a good answer surveys forms of assessing body temperature prior to 1743 (see time points above).
In Ring's work of 2006, he stated that, Florentine and Venetian glassblowers in Italy could measure body temperature using variations of Ferdinand II's early liquid thermometer which were tied onto the body surface. "...The rising or falling of small beads or seeds within the fluid inside the container assessed the temperature of an object.." (in this case a human body).
Before that, Santorio's air thermometer has been described to assess body temperature.
Considering these two published sources, one could propose that perhaps thermometers in its various stages of development, prior to the current modern form, have been used to measure body temperature.
However, it is not clear to me from these articles as to the extent that the use of the early thermometers to check/assess body temperature was widespread prior to the advent of the modern thermometer.
I would also presume that prior to the advent of any thermometers, even the early air-thermometers, the hand, especially a trained hand of doctors or nurses or other health providers, was the main form of assessing body temperature. This approach without using measurement devices also includes other aspects of clinical examination such as visual observation (e.g. change in skin colour, or sweating), or listening (e.g. increased heart or respiratory rate).
C. Latest developments in measuring body temperature
For added information, Ring has summarised the latest developments in the assessment of body temperature. These focus on how thermal imaging and other electronic devices have eclipsed the use of liquid thermometers.
Wright WF, Mackowiak PA. Origin, Evolution and Clinical Application of the Thermometer. Am J Med Sci. 2016 May;351(5):526-34
Wright WF. Early evolution of the thermometer and application to clinical medicine. J Therm Biol. 2016 Feb;56:18-30
Mackowiak PA, Worden G. Carl Reinhold August Wunderlich and the evolution of clinical thermometry. Clin Infect Dis. 1994 Mar;18(3):458-67
Barnett MK. The Development of Thermometry and the Temperature Concept. Osiris 1956 12:, 269-341
Ring E. F. J. The historical development of thermometry and thermal imaging in medicine. Journal of Medical Engineering & Technology. 2006. 30:4, 192-198
Ring E. F. J., Mcevoy H., Jung A., Zuber J., Machin G. New standards for devices used for the measurement of human body temperature. Journal of Medical Engineering & Technology. 2010. 34:4, 249-253