This feels very much like your sauna controller, either by physical or software limitations, simply can't deal with multiple button presses at once.
You had a rather nice approach here:
i thought about putting in a passive low-pass RC filter (i think it's just a capacitator and a resistor chained together?) on the line out from the "turn up" button, as that would solve "who gets there first", but then i wouldn't be able to press "turn up" as the circuitboard is busy listening to "turn on".
So, your idea is to add a low pass that basically delays your first turn up button press. Adding a delay: excellent idea!
So, for me, this looks a lot like you know damn better than the Sauna manufacturer what you want of your control panel :)
That, in turn means that you'd be the right one to design something that "pushes the right buttons in the right order" for you.
In the case of the turn-up button: It feels like a simple timer IC like the often-referred-to-yet-horrible 555 would actually do, if you drove e.g. a transistor with it that "shorts" the button.
The real question here becomes is what "makes the button act like a button", because that defines how you can simulate a button press.
Is it an opener, closer? Does it connect supply voltage to a pin that is otherwise tied to ground? Or the other way around? Or is it actually a clever button matrix where the microcontroller of the Sauna sequentially checks which pins are connected?
In the worst case, I'd still expect a small relay, driven by your logic (maybe through a transistor) in place of the original button to work. It's probably still easier to figure out how both contacts of the switches relate (i.e. which one has the higher voltage?) and just use transistors to replace the switches.
Now, to control these switch-replacements:
I mentioned the 555, as it can do something like an adjustable delay, or a periodic defined-length pulse. That way, you could emulate someone waiting e.g 10s after power has come on, and then pressing the "up" button ceaselessly every 0.5s for 0.1s or so.
I personally don't like the 555 very much, as more complicated circuits than what is described above often end up being very complicated and inaccurate. From my perspective, a simple microcontroller would do the job very well. If you have an Arduino, an Arduino-compatible board, or an mbed platform board (the simpler ones of these often can be had for but a handful or two of dollars), you might have a pretty easy time writing firmware that does exactly what you want now – and in the future. For example, one could imagine you add a digital temperature sensors (with I²C connectivity) in your sauna. Your controller could then query that sensor regularly, run a clever control loop and switch the heating on or off so that you always run at target temperature. Or you add a humidity sensor and a small pump that sprays water :D The sky is pretty much the limit as soon as your logic gets flexible!