Regulated multi-voltage PSU given a center-tapped transformer and a single regulator/zener
Nooooooo!

The transformer

Looks like it is a vintage cathode heating transformer? 3.15 x 2 = 6.3 (the usual heating voltage). Except in pretty rare cases, you can power the filaments directly (and older designs do). Most of lamps are pretty tolerant about +/- 10% or so.

The rectifier and the capacitors

These 4.4415V are more likely 4.0 if you are using Shottky diodes or 3.5-3.7 for the usual silicone rectifiers. Diodes have a forward drop voltage, remember. Doubled, it looks like 7.0-7.5V.

The 7806 regulator
These require at least 2.5V and better 3.0-3.5V (accounting for pulsations) more input voltage than the output, or they refuse to work. No 6.3V for your tubes.

No enough room for the resistor and zener regulator, either. The output of 7806 will be less than 5V here.


Otherwise, solving all of the above, you can simply ground the negative rail instead of the center tap of the transformer.

Edit: how to improve the situation

Lamps powered directly from the transformer (filaments are OK with AC).
Single Shottky diode rectifier and single capacitor. You'll get near 8V at the capacitor.
Either the same resistor + zener or a single 7805 or 78L05 and here you have 5V for the MCU.

Half the parts and all problems solved.
Edit 2:
Did you do something to regulate the anode power with respect to power fluctuations? If not, the heating not being absolutely stable may be way less of a concern since we are not talking about kW-range tube (these are picky and can go bad from either too high or too low heating). There are two other things to consider: there is some (capacitive) cross-talk between the heating circuit and the cathode. The heating transformer is center-grounded in order to get the cross-talk to self-cancel. ... or you will get hum. And second, you may as well still go with regulated DC heating, ground its negative and get rid of any hum, but keep in mind that all direct and even some indirect-heated tubes age faster when heated by DC (not sure if it applies to 12AX7). That's why if you power them with DC, you may want to create some means to switch its polarity once in a while.
All things considered, stick with AC heating.
If you need a common ground between the mcu-part and everything else, you are limited to using only one of the heating transformer legs. In this case, in order to get 5V, you may use voltage multiplier. You'll get 7.5-8V in respect to ground. Or two mulipliers, to keep everything absolutely symmetric.