Screens are not universal. There are physical differences in where and how the connecting cables are routed, what connectors are used, the electronics and interfacing, etc. Almost anything can be done if you have enough time, talent, and money. However, the chances are low that the two screens are interchangeable in the sense that all you need to figure out is how to physically attach the replacement screen and then plug in its cables.
As far as your two laptops, a better solution would be to buy a direct replacement screen if it is worth repairing the HP or put that money toward a new laptop. Stick a lightweight Linux on the Gateway and give it new life. There are many that will run well on limited resources in an old machine. Some are small and load entirely into memory (applications and all), and can knock your socks off with lightning performance on a tired old machine you thought was worthless.
This is not a direct answer to your question of how do I go about swapping the screens. That is unlikely to be practical, so this is an alternate approach.
The gateway also has a faulty motherboard and is prone to overheating, in addition to being way underpowered... The point is thst neither are worth the cost of fixing, but I have the time and don't mind fiddling. You're saying that assuming the connections are the same it's a basic switch - and if they're not? Is it undo able or just impractical? – Dovid Levine – 2014-10-13T16:51:19.930
If you are going to trash them anyway, there is nothing to lose by looking. However, the odds that one will be plug and play on the other are extremely low. Even if the cabling happens to be close enough and the connectors are the same type, there may be differences in the controls (like brightness/contrast), and in the interfacing. Unless you have the right tools, (and wiring diagrams), it would be very difficult to even replace the connectors. If everything physically matches, try it. If the connections are different, it will be impractical and uneconomical to modify. – fixer1234 – 2014-10-13T17:07:02.517