OP:
There are many places that sell refurbished or rebuilt long and short blocks, especially the 3.8L minivan engine used in the early generations of the JK/JKU/JKUR chassis from 2007-2011.
There are shops out there that will procure a used or remanufactured engine and swap it out in a day or two if arranged or scheduled in advance and ahead of time in most areas of the country.
Here in So. Cal for example, there are at least two or three salvage yards that have 5 or 6 of these engines pulled and ready for purchase to swap in. You can get them with or without accessories attached as your cost will vary depending on what the motor comes with and the mileage it had on the clock.
On the other hand there are several companies that specialize in used pulls and several more that specialize in reman engines. Companies like Jasper for example have remans ready and even include warranties on their remans.
I am not suggesting or representing any one company but there are ways to get an engine relatively fast for the swap unless maybe you lived in the most rural small community in BFE, I think you can find a shop and get an engine for the swap to be done in a timely manner of under a week even if they were painfully slow doing it.
There are many shops that are highly regarded if you want to travel to another state and visit one of these Jeep shops that will take care of your swap as needed also besides a general mechanics shop should you choose to go that route.
If you decide to do an engine swap to some other type of motor, you will be in for a LOT more money and time. Typically it can take 2 to 3 weeks at minimum for a HEMI, Diesel or GM LSx swap to be performed. Get your checkbook ready too because a lot of custom stuff will have to be added or done, especially if there are vehicle emissions testing in your area where your registered address/garage is located. Transmission swaps, wiring harnesses, emissions and custom parts abound when you swap engines.
My wife and I decided to sell our 2009 JKUR rather than swap the engine out to a V8 or diesel and buy a later Jeep JL series that came with the engine in it that we wanted from the factory. This was rather more appealing than the idea to keep the endless checkbook open for a power-trian swap to be done. It will keep our sanity in the long run especially considering we got what we originally paid for our JKUR over 8 years and 120,000 miles driving later! We can't complain ONE bit about that.
RR