First, consider how the circuit works. It's a Wheatstone bridge, although you might have to squint and turn your head to see it. Think about how "Tune" is a connection to ground through what should be a 50 ohm impedance. Here it is redrawn a bit:
simulate this circuit – Schematic created using CircuitLab
When the antenna is tuned, then R2 and R3 make a voltage divider, and R1 and the antenna make a similar voltage divider. As the input is driven by the transmitter, the nodes between each of these voltage dividers are at the same voltage (because they are each the same voltage divider), and thus there's no voltage across T1.
When the antenna isn't 50 ohms, then it is not the same voltage divider as R2 and R3, and so there is voltage across T1.
The bit with the other half of T1 and the LED is essentially an AM detector. Think of it like this:
simulate this circuit
D1 rectifies the input: when V1 is at its maximum the diode permits current to raise the voltage of node A by charging C1. When V1 swings back down the diode is reversed biased so the only path for current out of C1 is through R4 and the LED. So the voltage at A is approximately the peak voltage of the voltage source.
Of course the voltage source is actually the secondary of T1. The winding ratio of T1 helps to generate a higher voltage for the detector so it's better able to overcome the forward voltage of the diodes.
The selection of D1 isn't terribly critical. Anything that rectifies will work here. Even a razor blade could work. Ideally it will have a low forward voltage and a fast recovery time, but if it doesn't it just means it will be harder to light the LED, so you may not be able to tune as precisely. 1N34A is a germanium diode which has a lower forward voltage (around 0.3V) than silicon diodes (around 0.65V). Germanium diodes are an antique item these days. You might have better luck finding a Schottky diode, and it will probably work better too.
But if a 1N4148 is what you have and it works acceptably, then go with that. Nothing bad will happen.
You don't strictly need C1 either, but it might make the LED brighter. Without C1 the LED will only blink, but with C1 it can charge up and power the LED though the entire AC cycle. You can also think about how C1 lowers the AC impedance of the detector circuit, enabling it to draw more power and thus make the LED brighter.
Calculating R4 is pretty noncritical. You want something that's high enough to limit the current through the LED so it doesn't go up in smoke, but low enough that the LED is bright enough to see. Ideally you want the time constant of R4 and C1 to be long relative to the frequency so the capacitor doesn't discharge much between cycles. But the time constant shouldn't be so long the LED continues glowing long after you've reached a good tuning.
Now it should be apparent how you might adapt this circuit for use with a panel meter. Depending on the kind of meter you have and how sensitive it is, you might even forgo the transformer, and instead put the meter in place of T1's primary in series with a rectifying diode. Otherwise you might play with the winding ratio of T1 to make good use of the meter's range, and you can replace R4 and D2 with the meter.