Okay, when I took measures between the ends of the windings, I measured like 4 mV AC at both the Pink-Grey connection, as well as the Brown-Brown connection. Which is also strange.

I put a new fuse in before I started measuring tonight. Checked it for continuity as well, the 'old' fuse also checked out for continuity. The schematic asks for a 2 A and 500mA fuse, the TT kit came with two packages of each some 500mA and 2A fuses. However, there's only one fuse-terminal. Took the tubes out as well.

I turned the amp on for just a brief moment for a measurement at each of the points at a time, and to reduce chance of shocking myself accidently. But I'll leave the amp off for now and proceed with the method suggested. I did double, or even triple check the path of grey wires before I soldered them, cause it was a puzzle for sure... But with deducing I was quite confident I made right connections, grrrr... 

Still, the pilot light not turning on is a weird thing indeed... It beeps for continuity when I measure at the two terminals.

Okay, when I took measures between the ends of the windings, I measured like 4 mV AC at both the Pink-Grey connection, as well as the Brown-Brown connection. Which is also strange.

Not sure, what you did.
But must never directly connect two wires of the same winding, because that will cause a short.

No, I never did connect two wires of the same winding together. Only the brown-grey connection that asks for it to put in series on the primary side, see included photos earlier in the thread. The orange and white wire I cut-off, and put heatshrink on, individually isolated of course.

Can you measure the 230V mains voltage between the black and blue primary wires?

Dudeee, how could I've forgotten to measure that... Well, after a while of troubleshooting, sometimes you're just starting to overlook things.

I measured about 18v AC between the black and blue wire...  Hmm... Could it be the switch then or I got the wiring wrong there?!

Use your Ohmmeter.

[However,]

Then that's also giving me 18 ohm.

What's also confusing me is: The schematic indicates a 2A fuse in between the live from the power-jack and the switch and then a 500mA fuse between the Centre Taps and ground... However, Tube Town's lay-out only indicates one fuse with the 2A rating, where the 500mA should be according to the schematic (...if I understand it correctly, I don't know anymore if I do at this point....). But there never was a second fuse-holder provided with this kit, nor indicated on the lay-out?? Or on the TubeTown's own pictures of their finished kit, for that matter.

- Schematic: https://media.tubetown.net/cms/userfiles/downloads/tt-18w-schematic.pdf
- Lay-out: https://media.tubetown.net/cms/userfiles/downloads/tt-18w-layout.pdf

The HT fuse holder is on the layout picture bottom left.
This is for the 500mA use. HT= high tension = high voltage and means the plate supply.
The 2A mains fuse should be inside the IEC mains cable receptacle.
If you didn't know, find it and test it.

Where do you measure 18 Ohm?
I meant you should check the mains switch contacts with your Ohmmeter. 
It is possible you miswired the power switch.

I still suspect a mix up on the various grey wires.
The grey wire for the heaters (which goes in conjunction with the pink one) should be easily identifiable as it is a 1mm² whilst the other two greys are 0.5mm². I reviewed the provided pictures and it seems like this one is correctly hooked up.
For the two thin grey wire I see one going to the fuse, the other bundled up with the brown/orange/white.
I do not know how the Transformer looks like in terms of wiring exits, but usually Primary and Secondary side are separate, so one thick and one  thin grey wire should be on the side where black/brown/blue/white do exit the bell cover, the thick one is the heater, the thin one the start of the 2nd primary 120V winding.
The other thin grey should exit the other bell cover, together with the purple one's , and this is the HT CT (which then is fused via 500mA to GND).

The whole story shows again that building an amp is not a novice project and that there is certain basics in electrical engineering required unless someone is really keen on getting a Darwin Award presented.

Gruß
blues