Are there not apartment complexes in the UK?
Not in the way you are used to, no. There are not many buildings that have proper leasing offices like we're used to, just walking in and being shown around any open or show flats. In fact, I'd say most larger buildings that we could equate to looking like an "apartment complex" will most likely be owner occupied and rented out direct by a private landlord or via estate agent. This is not always the rule, as I know of a place in Sheffield that I saw online recently that is renting out units via their sales office.
Best way is to use a property search portal like already mentioned. These are sites where estate agents will submit their rentals, kind of a catch all. Keep in mind that these sites are often woefully out of date because of how quickly properties are snatched up, but at least it can direct you to an estate agent that may have similar properties. Right Move is my fave (great phone app too), but you can also use:
primelocation.co.uk
findaproperty.co.uk
vebra.com
To get more private landlords, (there's a filter to chose between private and agency) check out the following:
gumtree.com