When I've heard it used, 'look alive' means 'be on the look-out'/'Don't be "caught napping"'/[or, a more recent American* - ball-game based metaphor]'Heads up'.
A similar expression that does mean 'hurry up' is [the command] 'Look lively'. Sometimes the verb is ellipted: 'Come on men, lively now'. (This is a bit dated - you'll come across it chiefly in period dramas and war films.)
PS *When I was playing rugby (1966-'74) the traditional warning to look out for a high ball was just 'Heads'.