It has become apparent through email correspondence that some would be constructors of the receivers I've described are unsure as to some aspects of their construction.
So, I'm attempting to summarise on this page some important points.
First, the receivers in question:

https://www.cool386.com/6c4/6c4.htm
https://www.cool386.com/12at7/12at7.html
https://www.cool386.com/new12at7/new12at7.html
https://www.cool386.com/tradio/tradio.htm
https://www.cool386.com/amfm/amfm.htm

The main tuned circuit also applies to the pulse counting receiver:
http://www.cool386.com/pcrx/pcrx.html

The tuned circuit:
For the usual 88-108Mc/s coverage, this consists of 4 turns of 18 B&S tinned copper wire air cored, with a diameter of 10 mm. A drill bit or suitable felt pen makes a good former to wind the coil.
The associated tuning condenser needs to be 15pF for full band coverage. You can use other values of tuning condenser as I have done with some of the receivers; e.g. a 33pF in series with a 60pF variable will give correct tuning range, albeit with some extension of low frequency coverage.
Varicap diodes can be used as per the Model T Ford car radio; in which case you'll need to find suitable diodes, and provide the tuning supply from a well regulated source. Ideally this is based around a temperature compensated regulator IC specially made for the purpose. Typical types are TAA550 and ZTK33. Despite the way they are used, these devices are not zener diodes. Zener diodes on their own will cause tuning drift.
The best tuning device is an air spaced variable condenser. Failing that, the modern kind with plastic dielectric will work, although the losses are higher. Least to be preferred are varicap diodes with their low Q.
If you can get silver plated wire for the coil, it's worth using, but don't go out of your way to get it. I haven't tried the more common enamel or just bare copper wire in these circuits, but from experiments long ago these kinds of wire should work.

Those 15uH RF chokes:
A very critical part of the design. Get these wrong and the receiver will not oscillate across the whole band. The chokes I used were once sold by Dick Smith Electronics. They are no longer available from them. Don't ask for a part number because I had no idea who manufactured them. However, to describe, they are wound on a ferrite former and encased in blue heatshrink tubing. Both leads emerge at the one end, so they stand on end when mounted on a PCB.
There is a more common kind of choke which looks just like a 1/4W resistor. These do not work. I haven't measured their characteristics to determine why, but I suspect lack of Q.
The choke I now use and recommend is home made, not just because the original is hard to get, but because the performance is far superior.  The chokes consist of 75 cm of 25 B&S enamelled copper wire wound on a 6.3mm plastic form.

From top to bottom; 15uH DSE choke, 15uH axial choke (no good), homemade VHF choke (best performer), 1mH axial choke used in some pulse counting receivers, homemade choke for the Fremodyne.

Heater wiring:
The 12AT7 can have its two heaters connected in series to allow 12.6V operation, or in parallel for 6.3V. However, in these circuits this appears not to be a good thing to do.
I would prefer would be constructors to wire the 12AT7 for 6.3V operation. Pin 9 is to be earthed, and pins 4 and 5 connected together and fed from 6.3V. A 1000pF (.001uF) ceramic condenser is also connected from pins 4 and 5 to earth. Why is this all so critical when the 12AT7 has indirectly heated cathodes? The reason is due to the heater to cathode capacitance being very significant at VHF. So, the heaters are actually live at RF and therefore need to be bypassed to earth. Not doing so means the heater wiring will affect the operation, particularly in terms of the detector oscillating.
As for not recommending the heaters being wired in series, it would appear that despite bypassing pin 9, some RF is coupled from the detector back to the RF amplifier internally via the heaters. If you must persist with wiring the heater for 12.6V operation, bypass pin 9 to miminise interaction between the two heaters and earth the detector side of the heaters. Of course bypass the 12.6V supply to the heater. Again, I don't recommend this and it could be responsible for improper operation. For series heater circuits, the 12AT7 should be the last in the chain so that its heater is actually earthed.

Mechanical construction:
One constructor who was having problems had assembled the receiver simply by connecting the parts together in mid air. There was no chassis. Not only must lead length be kept as short as possible, the receiver must be built on a metal chassis. The tuning condenser must be mounted rigidly; not a trimmer capacitor suspended in mid air by wires.

Voltages:
Keep the B+ at 140-150V. Consumption is around  4ma including the RF amplifier. Heater requirements are 6.3V at 150 ma for 6C4 and 6.3V at 300 ma for the 12AT7 or 12AU7. Audio output from the 12AT7 detector plate (via the filtering components) is around 300mV. It is meant to be fed into a high impedance load; e.g.. 500K or 1M. Not solid state amplifiers with their low input impedance! If you must use a low impedance input, then use a FET input stage at the amplifier input or a cathode follower on the 12AT7 output.
Output is suitable for a two stage amplifier. Typical examples would be valves like 6BM8 or 6GW8, or where separate valves are used, 12AX7 and 6AQ5, 6J7 and 6V6 etc.
High impedance headphones can be driven straight from the filtered audio output at a level similar to a crystal set.

Valves:
With the commercially made 15uH chokes, 12AT7 must be used. With the home made chokes, performance is improved to the point where the lower gain valves 12AU7 and 6C4 (equivalent to half a 12AU7) will work. Interestingly, despite their lower gain, the detected audio output appears to be somewhat higher than the 12AT7 provides. Obviously if using 6C4's instead of the 12AT7, two are needed (assuming the RF amplifier stage is being retained).
Note that in varicap (and possibly plastic dielectric condenser) tuned receivers, the Q is reduced to the point where 12AT7 must be used; the 12AU7 won't work even with home made RF chokes.

As a final note, please read all the articles and look at all pics and circuits before thinking of constructing.