I wonder if I can install an American ceiling fan in a 240v country such as the UK by installing a step down transformer in the attic?

This may sound strange, but they do sell ceiling fans over here.
You have also got to factor in the difference in frequency, motors don't like it when the wrong frequency is generated.

Transformer will work. The frequency difference can always be accounted for by changing the voltage. (So instead of 120V provide 100V 50Hz)

For a single speed, even just a choke ballast will work. (But the phase shift capacitor must be changed accordingly)

 This not a Celing fan site as this for lighting
 Can poat this in off topic please.

  BTW 50 Hz would  turn a 60Hz fan a  lot slower. 
 

Running two fans in series or with a similar power incandescent lamp.

This not a Celing fan site as this for lighting
 Can poat this in off topic please.

  BTW 50 Hz would  turn a 60Hz fan a  lot slower. 
 

The fans usually have high slip, so the frequency does not have to be that strictly equal. So operating a 120V 60Hz fan on 50Hz and maybe 105V wont be any problem.
Ive seen many (shielded pole) induction motors rated for both 230V 50Hz and 120V 60Hz having just two windings to be connected in series for 230V and parallel for the 120V.
The capacitor shift ones needs different capacitor value (20% higher on 50Hz) though. But may as well start even with the lower one, it only taking longer to spin up.

I'll say this again, why would you go to all the hassle of converting the fan to run correctly when fans with the right specifications can be bought.?? 

Same reason as why run a lantern from the USA ? He wants that specific fan, not another one....

Well considering he's in the US, it's a silly question.

Its a what-if-would sorta question, but it makes sense

I was thinking about the idea of helping British and other European enthusiasts who prefer US style ceiling fans to look for ways to run them safely in their countries. For example, there is no such thing as a Hunter Original Ceiling fan deaigned for 240v as far as I know.

 120 volt 60 Hz fans  on 110 volt 50Hz the fan will run
  Thing is it will turn little slower.  It the Hz what control the
Speed on the induction motor. 

I heard that running inductive loads at low frequency increases the heat and so the life decreases, but for 10Hz less there shouldn't be any major problem.

The 10Hz may seem small, but it is 20%, so quite a lot.

For the fan:
The induction motors are primarily frequency controlled, but they exhibit what is called a slip - the difference between the real rotor rpm and the feed frequency. Now the slip depends on the ratio of (motor load) / (magnetic field strengt)^2.

With standard use, the motors are designed with very low slip (in the order of 2..4Hz), in order to limit the rotor cage losses.
But fans re usually designed as rather high slip (20..30Hz being the most common range with variable speed fans) and it is in fact the varying slip, what is used as a means to control the speed by the user (by varying the turns the winding has connected across tze mains).
So when you have 30 Hz slip at 60Hz, the 50Hz feed will cause 20% stronger magnetic field.
Which will reduce the slip by about 40%, so from 30Hz to about 20Hz.
So the rotor rpm at theend remains about the same...

If you cant find a fan design for a certain voltage area. The only options really is either make due with what there is, or perform a motor swap (or a blade swap) between the 120v and 240v fans. Most have blades that will fit onto most fan hubs.