Hello, axialhelp. Welcome to PF.
Axial tilt of the Earth is always the same. There's some variation due to
nutation and long term(thousands of years) perturbations, but in the time scale of human life or without the need for extreme precision, the angle of tilt remains 23,4° at all times.
This is the angle at which the Earth's axis is tilted away from the line perpendicular to the plane of its orbit.
What does change during the year, is the relative position of the Sun with respect to the direction the axis is pointing.
This causes the angle between the axis and the plane perpendicular to the direction to the Sun(the plane that bisects the Earth in two halves: the illuminated one and the one in shadow, or, day and night halves) to change from the maximum of 23,4° at Summer solstice, through 0° at equinoxes, to -23,4° at Winter solstice.
To find this angle at any given day of the year empirically, you need to do exactly what that BBC video told you to do.
If it's any time between the solstices, you'll get the angle between -23,4<α<23,4.
They even got such a result themselves, due to making measurements a few days before the summer solstice, as the lady tells you at the end of the video.
You could use sun path charts, like this one(
http://solardat.uoregon.edu/PolarSunChartProgram.html) to get the data you need for the calculations without leaving your room.
Alternatively, to a good approximation, you can just use this equation to get the angle you want:
α=23.4\sin({\frac{2\pi}{365}x}) where x is the day of the year counting from the spring equinox. When calculating remember that the value in the sine function is in radians, but the end result is in degrees.