Bought a new Ignition cylinder, vehicle will not start?

Did the new cylinder include new keys? Lincoln uses a version of the Ford PATS system for security. There is an RF tag inside the key that has to be recognized by the system to enable it to start and run. If the system doesn't recognize the key you get a no start. Fastest way to diagnose it is to use a factory level scan tool to access the PATS system and see if it has set any codes (Namely a B2103, B1600, B1601 or B1602) those would tell you where to look. B2103 = the transponder antenna isn't connected or is bad, B1600 = The key signal is not being received (usually see this one if the key doesn't have a chip) B1601 = The key is sending the wrong code or it isn't programmed into the system and B1602 which is that the key returned the wrong code format (there are a few different key blanks that fir that ignition but they use a different batch of chips in each one so that it is harder to make a key to steal the vehicle with.
On some of them a trick you can use to see if it is just that the key isn't recognized is to use aluminum foil to wrap the new key head very well, then hold the old key right tight against the new one and try to start the vehicle, if it starts you need to program in the new key(s) and it will be OK.
The best way to do that is to use a scan tool as Ford uses a tiered key system with a master key and spare keys. To program in master keys you need two working keys and the replacements, not easy to do if you have different keys as you basically insert the good keys, then the new key and have to do it in a time limit that negates trying to swap the cylinders in the middle. The spare key method let's you program in a new key with only one working key but requires you have the scan tool. In your case you will need to follow the process of all lost keys, for that you use the scan tool to access PATS, erase all of the old keys and program in the new ones.
Now if you are replacing the cylinder because it failed but have two keys you can take the cylinder from the truck to a locksmith with the new one and the original keys and they could key the replacement to fit the original keys, this has the advantage of needing no programming and the other locks will all still operate with the original keys.