Yes, it will get LTE but some early iphone 5 models will not get 3G/HSPA+.
You'd be stuck with 2G edge if you don't have LTE available.
Iphone 5 models manufactured after April 2013 will be compatible with all 3G and 4G bands for Tmobile.
Put the serial number into the applecare website and it will give you an idea of when the warranty started and if its before April 2013 then it doesn't have all frequencies available.
https://getsupport.apple.com/ProductSelector.action