I have just ordered a 650 W PSU for my computer which previously had a 300 W, in order to install a more powerful graphics card. However, I didn't check whether anything would blow up because it was too high - could there be any issues?
Also, what would I need to check to ensure the PSU would be compatible with all the other components?
There shouldn't be.
The wattage is the measure of how much power your PSU can deliver. The components will (unless they are faulty) only draw as much as they need.
Having a too low an output PSU will cause the PSU to fail in the first instance, which may cause other components to fail as a side effect.
In relation to your edit - there's not really a "compatibility" issue with power supplies. You can get converters (molex <> sata) and splitters if you need to connect older devices. The main thing to ensure is that the combined power consumption of all your devices (hard drives, motherboard, graphics card etc.) is less than the output of the PSU. With a 650W PSU should have a good buffer.
The "W" in 650W refers to the wattage not the voltage. The higher wattage will not cause any problems for your PC
There's no problem, you can have a PSU way more powerful than needed. In some case it's even recommended.
Yes, there will be issues: As sawdust already stated in a comment, you will most likely waste power with a too large psu since it will run in a less efficient part of it's range.
As the 80plus label tests only apply at 20% rated power upward, you might end up <20% and waste more then 20% of the power in the psu.
It will not damage anything, but higher power drain is definitly an issue. Often you can get away with a not too large, high quality psu - I've operated a very thirsty Phenom (125W TDP) together with a 80W TDP Radion 4830 on a 300W powersupply without any issues. GPU manufacturers often add a margin for low quality PSUs that don't deliver their rated power (or not enough on 12V), so you don't need to add an extra margin when you buy quality.
