Why do the radio and lighting work with the key turned, but not other things? For exactly the reasons RangeRoverHP and I just described! The fact that you just changed the starter is a HUGE red flag...
?You've recently changed the starter, which means you've disturbed the connection point for EVERYthing electrical in the vehicle.
?Suddenly, huge parts of the electrical system are without power.
If you want a more specific explanation of why some things work, and others don't, you need to take a look at the wiring diagram. Basically, there are 2 "main" connections at the starter. One of them (surprise!) supplies power to everything in the car that works with the ignition OFF. Since your headlights and other unswitched things (like the courtesy lights) are working, we know that this connection is OK.
The other connection is to the starter solenoid. Your starter works when you turn the key, so we know that this connection is ok, too. So let's start looking beyond the solenoid. It has two connections. One goes to the ignition switch (ok, because the starter goes when you turn the key), and the other goes to the alternator.
If power *was* reaching the alternator (from the starter solenoid), the dash lights would all come on when you turned the key. Since this doesn't happen, that substantially narrows the possibilities. No fuses or anything else are in the way. The charging/ignition light is on one wire, and the "bulb check" ecu (which turns on all the other warning lights) is on another, and both connect directly to the alternator. If you turn the key, and none come on, one of the following has happened:
1. the battery is dead (but we know it's ok, because the headlights are working)
2. there's a bad connection at the starter (but we know that's ok because the starter works when you turn the key)
3. there's a bad connection between the starter solenoid and the alternator
4. the ignition switch is fried (but again, the starter works when you turn the key)
5. the bulb check ecu & ignition warning bulb have BOTH failed simultaneously (pretty unlikely)
See where this is going? The plausible explanations are number 3 and possibly 4 - for the warning lights AND everything else that's not getting power.
You need to check these connections first. And if they seem ok, it's time to pull out a meter or test light and TEST them to see if they're getting power.
The solution to your problem will not be to pull the lower dash or the the steering column panel. For all these things to have stopped functioning means:
either there's a bad (or no) connection at a major distribution point (as detailed above), a partially fried ignition switch, or the most exciting alternative, which is the catastrophic failure of dozens of fuses and electronic components. Which seems most likely?