For scientific work, current meters are typically used at various points and depths across the cross section. This equipment would cost in a few thousand dollars.
But for rough estimates, you might try dispersing 10-20 oranges across the cross section, and record their individual and collective speed at some distance downstream, usually through a sinuous meander sequence or reach. Water velocity in a stream or river is variable with location, depth and some will be faster typically at thalweg, and some much slower, pools, eddies, etc. The surface waters where the oranges float may be faster, but some oranges will move to stronger current, and some delayed in slower currents. You might need binoculars to help locate oranges, and a stop watch.
If for for some reason, such as a stream gauging station, you know the river flow discharge rate in cubic feet per second, if you divide by the cross section area (in square feet, assuming you had that too), you would have the mean velocity.
There are various other other tools and meters, but measuring a river can be risky to dangerous if inexperienced or untrained. Doppler meters are most accurate, and suggested for most accurate in larger and more complex river crossings. This equipment costs probably above $25K today.
If you try wading smaller streams, don’t exceed the rule of ten for safety. The depth in feet times the velocity in feet per second should not exceed 10. A walking stick or wading rod can help. Safety ropes and floatation devices sometimes appropriate if wading.
I am not the pro here, but there is a method to measure discharge by addition of salt. For measurement, I think you simply need a stopwatch and a device to measure conductivity. If you search for the method („salt tracer“), I am sure you find helpful resources.
Hence you said 'improvised' I'm guessing standard instruments and methods are not possible.
Are you measuring a surface level velocity or at different or given depth?
If it's a stream or a small/narrow river, you can use the ping-pong method.
1. Get a ping-pong ball, metre tape, stop watch, at least one partner and personal safety equipment (boots)
2. Define a transect you wish to measure up to 10-15 metres from points A-B
3. Carefully lower the ping-pong ball without push or strong wind at point A in the direction of flow (of course) and have your partner stand at point B with the stop watch/clock
4. Your partner must keep an eye on the ball and hit the Start once you drop the ball on the water from point A (the person at point A may as well scream START! Once the ball is lowered on the surface water).
5. When the ball gets to point B, the person hits the Stop button of the watch and the time recorded in min:seconds depending on the calibration of the clock device.
6. Repeat this procedure up to 10 times and only consider closely related figures in the number of counts.
7. Go to lab, using simple physics formula for calculating velocity, m÷s, calculate using the mean distance(m) / mean time (s)
Distance is the space from point A-B e.g 0-10, 15-25.
If possible also sample the macrobenthos to provide information on the water velocity.
Limitations of the method
1. Resistance due to weight of the ping pong ball is neglected
If you want to measure water flow instead of water velocity, the salt dilution method can be used. The salt dilution method Principle The water's conductivity increases with increasing salinity. By adding a known amount of salt per unit of time, for example per second, and measure the water's conductivity before and after adding salt, you can get a measure of how much the salt is diluted, and thus the water flow. This method is of course best suited to small watercourses with small water flow. Saline solution can be dosed with a Mariotte bottle: https://en.wikipedia.org/wiki/Mariotte%27s_bottle One measures how much the conductivity changes due to added salt solution. When you know how fast the salt solution flows, and how much it is diluted, you get the water flow. The average water speed can be calculated by looking at how fast the water flows through a known cross-section of the river. M3/s /M2 =M/s