LEDs will emit steady light if given steady current. The question then becomes what kind of current waveform LEDs are driven with in a light assembly.
The exact answer depends on the circuit in the light. LEDs run on just a few volts. The much high line voltage has to be converted to the lower LED voltage somehow. Most likely, this will include the cheapest, dumbest, and most stripped down switching power supply possible.
I see other answers mention a transformer, but I think that is quite unlikely. There is no need for isolation in a sealed unit with no external connections other than the power line. Even if there is a transformer, it's not going to be fed directly from the line frequency. The extra savings in cost and size of a transformer that works at 10s to 100s of kHz far outweighs the cost of the components to produce that frequency.
Most likely, there is a full wave bridge to rectify the AC line voltage directly. That will then be chopped thru a inductor to drive the LEDs. Depending on how cheap the lamp is, it might chop at a fixed duty cycle, which would make the LED brightness vary with power voltage. Even a little current feedback would keep the LED current reasonably constant over most of the line cycle, perhaps dipping only briefly at the power line zero crossings. A small cap would reduce that, but caps cost money and take space, so may not be included.
LED lamps are made in large volumes, so serious manufacturers probably develop custom ICs just for this purpose. In such a case I'd expect at least some attempt at regulation, so the LED brightness will be largely constant, perhaps with short dips at the power line zero crossings.
However, all this is speculation. Why not just look? Put the lamp at the end of a extension cord and swing it around in a otherwise darkened room. Whether it flickers or not will be immediately obvious.
It kind of depends on where you are (UK?). Your information doesn't match exactly with my experience with fluorescent-lamp sizes. As I understand it, the number after the T in the designation is the diameter in 1/8ths of an inch (in the US anyway, I think elsewhere it is in mm). Typical old style sizes in the US are T8 and T12 which would be 25mm and 38mm respectively. The T5 series lamps would probably be the 16mm version you are referring to.
T5 lamps are electrically different from T8, although they are designed to be roughly the same Watts per unit length. T5 lamps can only be run from an electronic ballast. T5 lamps also produce more lumens per watt than T8, and run a little hotter.
T5 lamps are not compatible (physically or electrically) with T8 or T12 lamps, to make it work requires a conversion effort
Best Answer
I'm going to go out on a limb and say this question is valuable from the point of view of electronic design, as it pertains to some fundamental understanding on how fluorescent lights work.
Fluorescent lights work by accelerating electrons from the cathode to the anode in an almost-vacuum environment. In this vacuum is mercury vapour, and when the electron hits a mercury atom, that Hg atom goes into an excited state and outputs one or more photons of UV light upon decay. These UV photons then hit the phosphor-based coating on the inside of the glass tube, which converts these UV photons to visible white light.
So, in order to function, it is vitally important for these lights to have a lot of 'free' electrons available to shoot at the mercury. One way to make electrons more mobile and likely to shoot off the cathode is to heat it up, and this is what a so-called 'starter' circuit does: it is essentially nothing more than a high voltage generator and a heating coil. The heating coil heats up the electrode to mobilize the electrons and the high voltage generator (usually just a resonant LC pump) creates enough voltage for the initial 'spark' to ignite the bulb. Once electrons start flowing and the lamp is 'on', the gas inside the lamp looks more like a plasma and is very conductive, so neither the high voltage nor the addition of heat is necessary to keep it working. Hence, it's just a starter, once the bulb is on, it is shut down.
Old-style starters would keep trying to fire the bulb even when the electrodes were entirely spent. This means that that heating coil would be running until its filament would burn out. In a lot of cases this would mean the bulb has a higher power consumption after it's died.
Modern electronic starters 'give up' after a few tries when they detect that the bulb won't start. After that they use up no or almost no energy until power is cycled to the starter.