When to go
Best time
Sunrise, Blue Hour
Photograph from the Tower Bridge any month of the year.
I recommend going to this spot just before sunset during blue hour. Sundays are even better, as people sleep in on the weekends. The city lights will still be on, the River Thames will be still and if combined with a Neutral Density filter, you'll get some amazing reflections. 3 or 6 stops ND filter will be enough at that time of the day.
Field tips
If you're not able to bring a tripod, you can also use the bridge's ledge to steady your camera. Be careful not to drop your gear into the river!
To frame this particular shot, I used a wide-angle lens (35mm). This allowed me to incorporate an area called "More London," as well as focus on the city and avoid having the foreground (the bridge wall) out of focus.
The river Thames is always still in the morning (boats start to navigate on the rive around 6/7am), so this is another advantage of going to this spot early. You'll get an even smoother Thames at low tide. To check tides time, simply google 'London bridge tide times'.