Solid orange power button = dead cpu/motherboard.

According to the Service Manual for the T5500, the diagnostics pattern you indicated (Reserved, Visual Off state; LED code 1,2,4) should be "...accompanied  by a Solid Green Power light state. If the power light is not green, see Pre-POST Diagnostic Light Patterns." (It says this in teeny print right below POST Diagnostic Light Patterns.

Since your power button is solid amber, the POST Diagnostic Light Pattern 1,2,4 code doesn't apply because the PC appears to have failed the Pre-Post. Unfortunately there's no 1,2,4 code with amber listed as a Pre-Post error.

Could be power supply as speedstep said. But when was last time you replaced the motherboard battery (CR2032 3-volt coin cell, ~$2)? I'd replace that first before going any further, if nothing else than to rule it out for only a few dollars. And I'd reseat the RAM modules in their slots after you remove the old battery and press/hold power button for ~30 sec, before installing the fresh battery.

EDIT: Always unplug and press/hold power button for ~15 sec before opening the PC case.

Ah, yes, you're right. Sorry, I didn't see that.

I've never replaced the motherboard battery. A CR2032 3-volt coin cell battery is in the post, should be here in a few days.  I'll give that a go as well as reseating the RAM modules. 

I really appreciate your help and let you know the outcome.

Thanks

The details are always in fine print..!

Don't know where you are, but you probably could have bought a CR2032 battery at the nearest chemist, DIY store or Walmart, etc. if you have any of them nearby...

Don't forget to press/hold power button for ~30 sec after removing the old battery. Reseat the RAM modules before installing the fresh battery. Probably also a good time to chase all the dust bunnies off the heat sink, motherboard, fans, vents etc with some canned air. (A vacuum cleaner can create static electricity, so you don't want to do that.)

Post back and let us know how it goes...

Thanks to postings here I managed to fix mine. It would not boot. Steps I took. 

1. Reapplied thermal paste onto CPU.

2. Replaced battery.

Switched it on and showed the errors in the attached photo. Perhaps that was the previous issues because since then it has been working OK.