I would try the GPU myself, re-seat the thing and clear the fans and especially check the power cables going in to the card (if present).
I would then try starting the box with bare minimal peripherals attached, especially external HDD's (those things can cause all sorts of odd delays and issues at times I found).
If this gets to the OS screen I would give it some time to load up, then try update, then RESTART. Its important to understand that when you shut down and start up Win 10 (and 8 + 8.1) save a 'good state' and use that to start up but when you restart it starts up the old fashion discover everything way, so a restart is a good idea after you got it to load up properly... then use shutdown and start up normally when its all up and running.
If you are still having issues...
I would try loading up 1 RAM stick at a time (if you have multiple sticks), then try taking out none critical cards like sound cards and specialised video recording cards, special networks cards, extra USB cards... basically try get the PC to start up with bare minimal it needs, then add back one at a time to see where it tilts.
I am bit leery of advising this next part as I dont know much about it myself...
IIRC Win 8 ( + 8.1) and Win 10 use a different storage system, I remember needing to set something in the BIOS related to the HDD for Windows, it may be worth your time researching this, but I am not too confidant about this aspect as I dont have much experience with this myself.
My OWN upgrade gave me a few heart attacks to be honest, and it took a few reboots to get the PC stable but since its been fine except for one annoying issue, the network indicator at the start will often says the network is down or off or has issues yet it works fine, and after a restart the network can fail and require a shutdown and startup... minor issues if somewhat irritating.
I like the changes to the OS from Win 8.1 to 10 (after I went through and sorted the privacy issues
Hope you get the issue sorted.