It would seem that the achievement description should also read ' ... and complete the mission ' as at least 2 times My partner and I reached / and blew the horn within 4 minutes but did not get the achievement.
Before reaching the gate ensure You All have speed potions ( and ideally be wearing the potion share trinket ) and then before the gate blows up heal up.
It is then a matter of legging it to the Horn using Your potion of speeds at crucial times. We used ours as soon as the gate blew , on the roof top before the bridge and once on the run up the final stairs !
Our run time was 2 mins 56 seconds which included one stop to revive one person and several checks of boxes ( looking for more speed potions )... So there is plenty of time ! It goes without saying DO NOT stop to fight any rats / ogres just pass them all !
As soon as the 1st person reaches the horn location He should go up to the horn and be prepared to blow it ... You can not blow the horn until ALL players are in the horn room and should be aware that whilst blowing the horn if You are attacked this will stop You blowing the horn.
Then finish the mission / survive the Skaven attack and make it to the wagon and the achievement is yours !!