Many years ago, I rebuilt a clock radio that had a transformer like that. The transformer had blown when the clock was connected to 415 V.
As most of the power was taken from the lowest voltage for the LED supply, I found a transformer with a centre tapped secondary, and wired that up as in your diagram to the LED supply.
I then connected the transformer outputs to voltage multipliers, using electrolytic capacitors and diodes, to give the higher voltage for the clock electronics / radio. It was basically two of these:-
http://www.creative-science.org.uk/mult/2multa.gif
with one of each phase of the transformer secondary so that the output was still full-wave rectified. If you connect the anode of D1 to the opposite phase, not to ground, you get more voltage out.
For the high voltage AC, I used a separate, tiny transformer. It was something like on of these:-
http://uk.farnell.com/block/vb0-35-2-9/transformer-0-35va-2-x-9v/dp/1131609 but it had flying lead connections. You can now get toroidal transformers like this:-
http://uk.rs-online.com/web/p/toroidal-transformers/2786762/
It is highly unlikely that the 50/60 Hz input has to go negative for the clock to count the frequency. You could just have a pull-up to the +ve supply, and a transistor to ground switched by the ac output of the transformer to give you a waveform that would be fine for timing. That would avoid the need for a second transformer.
As an alternative, find a transformer that will directly supply the clock electronics, the radio and the 50/60 Hz input, and then use a buck regulator to provide the lower voltage for the LEDs.
Another alternative would be to find a toroidal transformer for the clock electronics etc, and make sure that it is one with a hole in the middle. Then wind some turns of wire around through the middle to give you the voltage you need for the LEDs. I guess you will get 0.2 V per turn, but you will have to measure it. If you are taking as significant proportion of the transformer power from the extra winding, make sure that the turns are spread evenly around the transformer.
You can full-wave rectify with schottky diodes to minimise the number of turns that you need.