Yes, most new smartphones have this capability. Take the iPhone 5S for example. (https://www.apple.com/iphone/LTE/)
These are the supported LTE bands:
1 (2100 MHz), 2 (1900 MHz), 3 (1800 MHz), 4 (AWS), 5 (850 MHz), 8 (900 MHz),13 (700c MHz), 17 (700b MHz), 19 (800 MHz), 20 (800 DD),25 (1900 MHz)
700/AWS are the main 4G bands in the American ITU region
800/1800MHz are the main deployed bands in Europe/African ITU region
1800/2100MHz are the main deployed bands in the Asian/Pacific ITU region (note that APT 700MHz is different to the USA's mongrel of a 700MHz band)
Other bands (e.g. 900MHz) are only used very rarely (in this case one operator in Sweden and one in Czech Republic) but also are supported.
Your daughter's main problem will be:
a) whether her UK network has deployed 4G where she is (though in the middle of Edinburgh or Glasgow she should be fine). You will find that due to better 3G networks, Europe is lagging behind the US in 4G coverage.
b) the lack of 4G international roaming (not many operators let you roam onto 4G networks)
c) the cost of 4G international roaming (if allowed) would be prohibitive