3mm and 5mm LED

Note that the paragraph that accompanies the circuit specifies the LED type as a flashing LED. It says that a normal LED will not work. The paragraph does state that the LED in the picture is 3mm, so I can understand the confusion. However, a flashing LED has a small circuit built in to make it self-flashing.

This circuit needs that internal circuitry of the flashing LED to drive the NPN transistor.

In summary, if you can find a 5mm flashing LED, you can go ahead and use it! Just make sure it is a flashing LED though! A quick Google of flashing LEDs will give you numerous results with different sizes.

Any size LED will work - the warning is that it needs to be a flasher type LED, since it relies on the current drawn by the flashing LED to drive the transistor switching the other one.

Nothing in that says it needs to be 3mm. However it does say an ORDINARY LED will not work (You need a flashing LED).