EDIT and conclusion:
Thank you for this amazing response! I was led to reflect on this matter because I am translating some 16th Century Dutch and German acounts of the sleds/sledges/sleighs used by the Samoyeds, and I was wondering what the most appropriate term was. I am English and would most naturally use the word 'sledge' for such objects; dictionaries confirm that 'sled' is 'chiefly US and Canada', while not being an 'Americanism' in the sense of sounding odd or affected when used by a British person. All three words have the same basic etymology, being derived by different routes from Germanic words for sliding.
It would seem that 'sled' has maintained itself as the usual word for such objects in North America, and 'sledge' in Britain, while 'sleigh' generally has narrower connotations everywhere, being specifically associated with Santa Claus, reindeers and that kind of thing. I doubt that the latter was so much the case fifty years ago; Fowler indeed speculated in the 1930's that 'sleigh' would probably displace the other terms. Usage in Australia/NZ seems caught between British and American as so often.
With regard to 'sled' and 'sledge', however, there interesting differences in emphasis. Some North Americans find 'sledge' strange and unfamiliar, while others might use it alonside 'sled'; and likewise, some British people might use 'sled' while it strikes others as being more distinctly American. Furthermore, some find a difference in meaning between the two words, finding 'sled' more appropriate for a lighter and 'sledge' for a heavier vehicle. I felt this myself when wondering what to call the Samoyed vehicles, 'sled' seemed positively inappropriate for reindeer-drawn vehicles which could be used for transporting quite heavy loads. Why this should be, I find hard to work out, since the words are basically the same. Is it something in the sound of the two different words? So: 'sledge' it must be, not only because I am writing in English English but also because it sounds more fitting for me for the heavier vehicle. To be sure, because these are reindeer-drawn, perhaps I should say 'sleigh', but that conjures up associations of jingle-bells and all that, which would be out of place.