Basia - if you don't have time in Belize City, don't fret, you'll find a store in PG that will still be able to sell you a cheap cell phone/sim card (Belmopan for sure).
Answering both your question and Lan's, there are certain phones that you can bring here, and as long as they can be "unlocked," you can then buy a SIM card here and use right away. Check with the cell phone manufacturer to find out if that is possible.
When I first arrived in Belize, even though I had told Verizon I was coming here, I could not get my phone to work. Furthermore, they wanted to charge me $2.81 usd per MINUTE!!! to call the States, Yikes!! I just had a relocation client move here a month ago, and that rate has not changed.
As far as I know, not even 3G is supported here, so I'm sure 4G is not in the immediate future.
In terms of more cell phone providers, frankly, I don't think another one is in the immediate future. BTL is government owned for one, and they probably don't want a lot more competition. And with such a small population, not sure the market share is there for another company to make the investment.
Only time will tell though - never say never.
Hope that's been helpful.