In addition to what Clive has said, if you told me (I speak American English) that Sally was wearing a jumper, I would understand that Sally was wearing a sort of sleeveless dress. Something like
'Jumper' is one of those words (compare: braces, vest, tap) that have slightly different meanings in AmEng and BrEng.
All three are acceptable (and common) in BrEng, and most people would say "Put on a jumper" meaning "Put on a long-sleeved woollen or cotton top". Pullover is less common these days (as is its abbreviation, "pully"), and to me sounds more Northern. Sweater sounds
Regarding the examples Yankee gave of the American use of "jumper" -- I would only call a sleeveless dress a jumper if it was intended to be worn over a blouse or shirt, as the second and third of the illustrations show. I would call the first one a sundress. (Though I suppose you could wear a sundress as a jumper if you wanted to.)
Yes, I would also expect a jumper to be worn with a blouse (or possibly a sweater) underneath. However, the manufacturer of the dress in the first picture is advertising it specifically as a jumper that can be worn either with or without a shirt underneath.