My guess is the computer is not running actually even with the fans running.
I would first unplug the power and hang on to the metal to relieve static many times as you do the following. You probably need a new cmos battery or check your battery has 3.3 v dc or what is written on the battery. Then pull the ram and reseat it, blow out the contact area, then remove the CPU cooler and re-seat the processor, we do not know what processor it is but the old ones have a lever to release the processor lift this and sort of carefully jiggle the processor, if a different processor the same idea if you can pick it out a bit with fingernails and let it back ( we are thinking in the move things were shook up some like a heavy bump) take your time and be cautious lock the processor back. the grease pad should be ok, I usually dab at it with a finger then when putting the cooling fan back sort of squiggle it on top of the processor, if the computer runs then think about going to get some new grease (it does not need much as it is just filling in the depressions between the processor and cooling fan plate so there is no voids in the contact where the heat can build up and not transfer the heat to the cooler.
Same goes for the power supply connection on the motherboard and hard drive connections.
You get the idea I am sure, to just re-seat all connections on the hardware.
Dust the inside and fans to assure of cooling, then hook up your external connections for the video and keyboard and mouse and give it a go. If the computer is actually running you should hear the hard drives whine up then pop goes your video we hope.
With all this you are not out any money yet and the hard bump in shipping still could have broken a trace on the motherboard so you are hooped. Oh forgot for you to have a good look at your capacitors on the board for bulging tops or other damage. I have had computers running then stop for a reset of upgrade and it is dead.
Good luck and please post how you are making out.