Do transformers use Aluminium windings?
Yes. Aluminium is used for transformer windings and for other wiring use. As you note, Aluminium has higher resistivity to Copper so needs a larger wire size than Copper to achieve the same number of turns OR in a given winding space Aluminium windings will have a larger resistance than if Copper is used. 

Aluminium generally takes more care in construction than copper, but when properly implemented no significant problems are caused by using Al rsther than Cu.  

Where winding volume is not an issue or where the higher winding resistance is acceptable, Aluminium can be substantially cheaper than Copper for the same application.  

Here is an excellent discussion of the pros and cons. This goes into the various differences, their effects on design and how the effects can be made irrelevant or insignificant by good design. 

They note that 


In North America, aluminum is the predominant choice of winding material for low-voltage, dry-type transformers larger than 15 kilovolt-amperes(kVA). In most other areas of the world, copper is the predominant winding material. The primary reason for choosing aluminum windings is its lower initial cost. The cost of copper base metal has historically proven to be much more volatile than the cost of aluminum, so that the purchase price of copper conductor generally is the most expensive choice. Also, because aluminum has greater malleability and is easier to weld, it is the lower-cost manufacturing choice. However, reliable aluminum connections require more discipline and expertise on the part of transformer installers than that needed for copper connections.  


They also provide this comparison table: (reformatted)

Table 1: Common Reasons for Winding Material Choice for Low Voltage Dry-Type Transformers


FALSE: Aluminum-wound transformer terminations are incompatible with copper line and load cables.  
TRUE: Properly terminating line and load connections is more difficult for aluminum-wound transformers.   
FALSE: Line and load connections to copper-wound transformers are more reliable than those to aluminum-wound transformers.        
TRUE: Aluminum wound transformers are lighter in weight than copper wound equivalents.         
FALSE: Copper-wound low voltage transformers are better for "high-impact" loads because copper has higher tensile strength than aluminum.     
FALSE: Aluminum-wound transformers have higher losses because copper is a better conductor.       
FALSE: Aluminum-wound transformers have higher hot-spot temperatures because copper is a better thermal conductor than aluminum.      


Some of those need to be thought about. While Al is a notionally inferior material cost and resistiovity wise, the effects of the differences can in many cases be eliminated by proper design.