Excellent!
The battery is low, but doesn't seem collapsed. (95% confidence?)
Low is expected anyhow since it is undercharging.... (99% "not the battery" confidence.)
So it's an alternator problem - unless connection problem.... (fuse-links etc)
You mentioned no charge lamp.
If it's a single-wire alternator, can you disconnect the D+ wire (at the alternator) and run a lamp from D+ to +12V (battery) to see if it lights? Ignition can be off. The bulb should be (12V) 2W or 3W, else 5W, or maybe 10W - but no larger.
If it lights, can you keep it connected, start the engine and see if it extinguishes.
Please do not allow hands or ties or clothing to be caught in any moving part, fan belt, fan blade, alternator fan or pulley etc - it can hurt a lot, and kill a lot too!
If it does extinguish, check voltages at the alternator output (aka B or B+, should be 13.6 - 14.4V depending on RPM), and at the battery +12V (which should be close or the same as the alt's B+).
Heed my former plea - ie WARNING!
If the alt B+ varies from the battery+, you have a bad connection or blown fuse between B+ and Batt+ (aka B+).
If the D+ lamp doesn't initially glow/light, the regulator is faulty. It it doesn't extinguish after starting, the alternator or regulator is faulty or you have a broken fan belt, or missing pulleys etc. Or started the wrong engine.
If it isn't above 13.6 with RPM and loads (eg, headlights), the if not a slipping fan belt, the alt/reg is faulty. (Worn brushes, blown power diodes, etc.)
If it's a 2 or 3 wire alternator (regulator), the same testing applies, but you need to connect the 2W-3W test light to the D+ which they usually label as "L" (charge Lamp) as opposed to S for Sense or IGN, and I for IGN.
The L circuit should handle the extra "not big" test lamp for battery+ - ie, a 12V 2W or 3W bulb.
Note - although LEDs will illuminate like a bulb, they may not pass sufficient tickle current, and it can be hard to tell if they extinguished...
[FYI.... Because most alternator D+ or L outputs are say 1.2V under the +12V supply which is enough to light a LED as well as cause the charge lamp and various tested dash bulbs to glow dimly (at night) - and hence why modern cars often switch a relay to switch or power the tested dash lights instead. ]
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Did other lamps normally come on with ignition but engine NOT started - which then extinguished with the charge lamp? This does not include oil-pressure lamps (nor handbrake & hi-beam etc), but things like "brake-fault" of low fuel (for EFI) lamps that are tested upon IGN-on.
And do they still come on?
Of they do, it suggests a blown charge lamp (rather than a faulty alternator).
Many (most?) alternators must have a charge lamp to ensure that they begin to charge. Without it, they may charge, but often not - especially after a few hours since last charging.
Once initiated, the alternator may not need the charge lamp. (It depends on the model etc.)
And for older cars that had "tested bulbs" when the system was not charging, the tested bulbs could also provide the trickle or tickle current provided via the charge lamp to initiate charging.
But later cars use relays or other circuits for such tested bulbs and will not initiate charging. (IE - they are not connected via diodes to the charge lamp circuit from the alternator/regulator.)
There you go.
Short and sweet.