Hello MarkNash
I am having the same issue I believe. The laptop will power on (fans/cdrom power is evident). all three "LOCK" keyboard leds flash briefly then go off, and then nothing. I have taken everything apart multiple times, re-seated the RAM, cleaned contacts for pertinent components.
What I believe to be the issue, is a bad solder on the video chip. Since the Video chip solder is defective, it will cause a no POST issue.
First, there is a workaround to this, if you believe this to be your issue. I have my Vaio to work by simply turning it on and leaving it on for an extended period of time. Eventually, the mobo will warm up and can reconnect and broken solder, and simply turning the laptop off and then back on quickly and it should work. Time may vary, but it has generally taken anywhere from 1-4 hours before the power cycle will work.
A solution would be to have a reflow or reball done on the video chip/motherboard, which is basically re-soldering the video chip to the motherboard to get a permanent, lasting connection. In my experience, it is hard to find someone to do this though (I am yet to find someone who has done it, or is willing to do it in my area).
If you are unfamiliar with this process yourself, you could attempt to fix it by using a heat-gun on the video chip, but if you aren't careful you could easily damage other components or the video chip itself.