The power supply is as important as the rest of the amplifier since it supplies the power you are listening to. If the power is bad, the sound is bad.
There are people who insist to build the power supply the same way Williamson did because it sounds better they say. If audible at all, it is a matter of taste. The supply with a rectifier tube has a higher inner resistance since the power transformers that are made to work with rectifier tubes are made with a fairly high dc resistance in the copper of the secondary because the specs of the rectifier tubes clearly states they expect that resistance in the order of 100 Ohms in order to survive. Transformers made for solid state rectifiers have a dc resistance in the secondary as low as 2 or 3 Ohms. It is not impossible this difference in dc resistance results in the difference in sound reproduction that some claim to hear. So if you use solid state, try some resistance in the positive lead especially if the plate voltage is a bit high. The 450volts Williamson uses is a bit to high for electrolyte caps that are rated 450WV. 425 volts is saver. One must expect the mains voltage to swing 5% from the rated value! For 425v using a bridge rectifier you need a transformer the has a secondary voltage of about 310v. Using a rectifier tube you may need two times 350v to its plates.
If one can afford an extra regulated power supply for the input stage this is recommended! In a stereo amp two of those stages draw only 19 mA. so a small extra transformer will do.
Chokes are rather difficult to get nowadays and when you can buy them in specialized audio shops they are incredibly expensive as if those things were something special. Fortunately you can make them yourself from ordinary transformers with a E-I core lamination. Remove the lamination, put all the coils in series ( watch the direction of the windings !) and replace the core material so that the E 's are on one side and the I 's on the other. Allow an airgap between the legs of the E 's an the I 's by putting a thin strip of polyester in between. Test your choke with 50 or 60 cycles from a mains transformer at the current you want to use it for and refine it. Compare it with a choke that you know is good. Put it in a neat metal enclosure to make it look expensive.
In the Williamson design the reservoir caps are rated only 8 mfd. Still he claims his amp to have a signal to noise ratio of 100dB. In his days electrolytes were very expensive and chokes are a great aid in smoothing H.T.! Nowadays we rather buy a couple of 100mfd"s in reservoir and smoothing caps. Replace any cap that you think is to small for any value you think is better but remember that rectifier tubes don't like to see a capacity over 32mfd!
Some people will heat the tubes with ac others with dc. It doesn't make that much difference. Williamson used ac and achieved the aforementioned 100dB. If you use dc consider putting the heaters in series for 12.6v, you need only half the current and the loss in the rectifier is much less. In terms of ac you need aprox 10 volts. If you end up a little higher then 12.6 ac, build in some resistance, this prolonques the live of the heaters. The exact 6.3 or 12.6 volts at the rated mains voltage is very important for the livecycle of the tubes.
Since the cathode voltage of the phase splitter tube is about 100 volts you may consider to superimpose some 50 volts to the heater circuit to reduce the chances of a breakdown of the tube. If you heat with ac this also reduces the heater hum since the heater at this voltagelevel will not emit electrons. You can take the 50 volts from resistors between ground and HT that total some 200kOhms that you need anyway to discharge the reservoir caps. Try 22K and 200K.
When wiring the heaters don't forget that the current becomes enormous if you connect all the heaters in paralel. Especialy if you plan to use the EL34 with 1,6amps heater current each!