Solid State Dial lamp Replacement? Yes or No?

If your radio uses a power transformer a solid state LED replacement for the pilot light should work OK  But check the voltage being fed to the bulb.

 If your radio does not use a power transformer, DO NOT use a solid state replacement. The reason is that most rectifier tubes in the AC/DC radios have a tap on the filament of the rectifier tube.
 This tap generally is where the pilot lamp is connected. It is also PART OF THE FILAMENT CIRCUIT FOR THE RECTIFIER.

 The rectifier filament is designed to have that lamp in circuit. Just like having a burned out pilot lamp will shorten the life of your rectifier, putting in a LED one will cause a voltage & current imbalance. It may even cause the LED replacement to be damaged, or the rectifier.

Also keep in mind that when your radio was designed 110V was the standard. With line voltages becoming higher comes an increase in voltage on the filaments. I have seen voltages here between 120V and 125V now. That is a 10-15V increase. Accordingly the B+ will also be higher. Using a tapped isolation transformer on the input of the radio will stop problems.