That's interesting. Most of the on-line dictionaries are American but I found that M-W recognises the British pronunciation, and apparently it is heard in Canada too. Don't know what they say up top in the antipodes. This is the M-W entry and I have included the etymology for the heck of it:

Main Entry: 1 shine
Pronunciation: 'shIn
Function: verb
Inflected Form(s): shone /'shOn, esp Canadian and British 'shän/; or shined; shin·ing
Etymology: Middle English, from Old English scInan; akin to Old High German skInan to shine and perhaps to Greek skia shadow