What wheels can I buy that fit this Shimano UG 5 speed freewheel? I am refurbishing an old 70s/80s bicycle, I have everything ready but the wheels, what set of wheels should I purchase that will fit this freewheel from Shimano?
A bicycle of that era will likely need 27" wheels and 126mm rear spacing.

To confirm this, you could put a quick release or other straight small object through the dropouts (as if it were an axle) and measure the distance to the brake pads. A 27" is going to be around 630mm/2 = 315mm in radius to where the tire sits, so the brake pads should be able to extend a little bit closer to the dropouts than this (this is not exactly a way to determine what size wheels the frame was designed for, but just determine that they will work). Depending on exactly when it is from in that era and the quality of the bike, its possible that it would use 700c wheels (which have a very similar 622mm diameter) but very unlikely.

The spacing is the distance inside the dropouts:



from http://farm3.static.flickr.com/2061/1819835696_836e18f8ea_o.jpg

The rear should measure 126mm I would expect, and the front should almost certainly be 100mm (any wheelset that meets the specifications for the rear should also fit the front)