i) 0-15 seconds: no hum.
ii) 15-20 seconds: hum quickly builds up to quite high levels.
iii) 20-35 seconds: the hum gradually subsides to zero level (not audible anymore). I wish it stayed like that forever.
iv) 35-45 seconds: the hum slightly builds up again and stabilizes at a low but well audible level (~7mV pk-pk at speaker terminals, see first scope screen-shot in the OP).
Investigation and finding problem need take this timetable to account.
This is perhaps miss but least good to think if....
You told these EL84 are matched pair. But
do they have also matched filaments? (personally I suspect that matching do not include filaments) Very different cathode temperature also may lead to some imbalance. And this your timetable about hum.... at least not completely incompatible with cathode heating related residue imbalance. (im not sure if you can also watch it with eyes looking glow color when you turn it on, in worst case you may see other EL glow fast bright and after then decay and other EL filament brightness rise (if difference is big). )
Least check it, connect two meters (or scope channels) to record voltages over both filaments when it start and warming. I suspect that in start from gold they run very different and even other one may go totally overvoltage ( in worst case) until also other filament warms up in worst case. As we also know when filament warms up its resistance drop and then other get low power... and again think your time table.
Even if this is not any part of problem in this case but this kind of design is not "best possible" at all.
As someone have previously noted, transformers magnetic fields can also be source of problems, this is why we many times see that specially E type transformers have weird positions and angles. If transaformers are toroidal then it is more easy.
Then, filament current is also near 0.7A so this need keep away from sensitive signals, so it need least be twisted pair and better is far away from sensitive parts as also someone have previously noted.
Preamp filaments are DC and end amp is AC. But why designer have selected series mode for drive filaments. It can do, specially for 12AX7 but it is not necessary and with EL84 end tubes this method is - imho, bad.
World is full of good and top class and state of art tube amplifiers. Why this designer has drawn his own magpie's nest. It is intersting if can see how it have mechanically assembled and how these all GND's are connected.