In the United States, as far as I know, they are pronounced the same, and both of them rhyme with sun and fun. This is certainly true in the region where I grew up, the Northeast, and I'm not aware of any regional variation although there could certainly be some.
The desk dictionary I use, American Heritage Dictionary 3rd Edition, the word "one" is shown with a single pronunciation "wŭn," same "u" sound as in the word "cut." "Won," entry #3, "past tense and past participle of 'win,'" is also shown with only one pronunciation, the same, "wŭn." (Entry #1 is an archaic meaning I didn't know, "to dwell or abide," and entry #2 is a unit of Korean currency).
From what Su.Ki says, I gather this is not true in the UK.