In the OP's schematic pin 2 of the NE555 should connect to pin 6 not to +V. This error is not present in the Elektor schematic below.
wow. all that shaping of the square wave, you could just take the pin2/pin6 output and buffer it for a sawtooth-ish output.
I think the filter is intended to give a nicer sine output than just the exponential output across the cap. The advantage of *starting* with the exponential output is the level is automatically set by the NE555 regardless of the frequency pot setting.
To me the circuit around C6 looks wrong. I think there should be another 10k resistor between C4 and C6. Another options could also pull C6; which now seems more likely - see below.
The problem with making the frequency variable is the level of the sine output (and triangle and exponential outputs for that matter) will decrease with frequency.
Replaced R1 with a 1M pot and inlined the 10k resistor with that as without it would cut out if the frequency got too high, presumably outside the spec for the chip or something like that.
You would do better making R2 variable so the output remains more or less square when you adjust the frequency. Making R1 variable means the shape of the waveform changes when the frequency is adjusted: the output high time varies and the output low time is fixed.
For R1 variable, putting 10k in series with the pot is necessary to ensure the oscillator can oscillate. For R2 variable you don't need the 10k but it is very desirable because without it the frequency will increase rapidly at the end in an impractical manner - 10k to 50k is a good choice.
Looks like the original Elektor circuit has the sine filter error as well:
(Need a resistor between C7 and C4, could also just pull C4.
But oddly enough the kit PCB seems to have C4 loaded, like the schematic.)
https://worldradiohistory.com/hd2/IDX-UK/Technology/Technology-Modern/Archive-Elektor-IDX/IDX/00s/Elektor-2000-12-OCR-Page-0097.pdf
Seems like there's a few problems with that circuit. Some tentative fixes.
The idea was to work out what the circuit was based on the sine output level in the Elektor (Dec 2000) article.

I'm seeing about 0.7V p-p on the square output, 2.5V p-p on the exponential output, 0.7Vp-p on the triangle output.
With the square output the buffer is being totally cut-off on the negative swing because the 1k on the divider is killing the bias on the output buffer. The same thing isn't allowed to happen on the exponential and triangle waveforms as that would chop off the negative tips of the waveform. Luckily as you progress down the 10ks it finds a new bias point which keeps the output buffer biased enough not to clip.