You are well informed to discover that your laptop indeed will support the higher resolution screen. The LED cable is indeed supported across all versions of the t510 and 510i for the resolutions from 1366x768, 1600x900, and 1920x1080. Make sure you do have the correct LCD cable, which you will see directly on the cable as the FRU #: 75Y5557
Please do not confuse this with the LED cable, that connects the webcam and rear panel LED.
Here is a website you might want to checkout, which will detail your parts as well as the supported models:
You might notice that there is also a multi-touch 1080p panel that would be compatible, but you would be better just buying the whole assembly on that one, since you would have to buy a different set of hinges, rear panel, front bezel, and screen.
Finally, the part that you do not want to see. Generally I recommend sticking with the same type of screen laptops are shipped with. The higher resolutions tend to put a greater strain on the graphics card and your wallet, and unfortunately I could not find anywhere that could tell me exactly what Intel HD GPU they integrated on the motherboard; but so long as you are running the Intel HD 3000 or 4000 series GPU, you will be just fine with the 1080p screen.
I also do not have a t510 that I could use to make a guide on how to replace the screen, but having worked on them before, it pretty easy.
1. Remove the screw covers right next to the hinges and unscrew the two Phillips head screws to free the front bezel.
2. The rest of the bezel is snapped in, so you should be able to pull the bezel off starting from the bottom-center of the screen and working your way outwards and upwards.
3. Remove the 4 screws holding the hinges to the back panel, two at the bottom near the hinges and two at the top corners.
4. There are 3-4 more screws holding the screen to the hinges on each side. Simply unscrew those last 6-8 screw and the screen will be ready to replace.
Last but not least, here are some links to LCD screen vendors that I have been using for a couple years that you might also like!
15.6" HD+ LCD Screen 1600x900
15.6" FHD LCD Screen 1920x1080