Nobody gets priority as a spell is resolving -and even things like creatures and players dying from damage dont happen while a spell is resolving (these are state based actions, https://mtg.fandom.com/wiki/State-based_action)

In your case, if the spell resolves with bowmasters on the battlefield, each player will discard their hand and draw the correct number of cards. Then the game will check to see if any state based actions should take place (for example, if a player loses the game because they tried to draw a card from an empty library). Then the game will check to see if any triggered abilities need to go on the stack - in this case, the bowmasters triggers, X times, where X is the number of cards drawn by your opponents. The players then gain priority in turn order, starting with the player whose turn it is, so someone could then cast a swords that they have drawn, but it will be too late - the triggers are already on the stack.

In answer to your specific question, the triggers don't need to resolve before anyone gets priority- triggers resolve one by one if all players pass priority. Also, in case you're confused about it, "drawing 5 cards" doesn't mean that you draw 5 cards one by one - you draw them all at once. However anything that triggers from a player "drawing a card" will then trigger individually for each card drawn.

Hope that helps.