45 days is the worst case senario. Workers bees will forage, queen or no queen. In a walkaway split the population might decline a bit before the queen hatches, mates, and starts laying. But there are several factors in the beekeepers favor.
The unhatched brood in the frames used to create the split, the ability to add additional frames of brood, shaking in more workeds, or feeding the hive. The workers will continue to forage for nectar and pollen so that there is a store of both by the time the queen is ready to begin laying.
They will consume food but probably not as much as they gather so the food situatiion is a plus not a negative.
The negative is the same as you'd get with a package as far as decline of ppopulation is concerned and in a package the queen is probably idle the 1st week anyway and then in very limited way depending on how fast comb is built. In comb building only a portion can be used for brood as it is still necessary to set stores aside and much of the new comb is used for that.
So look at a split much like a package, the difference being a trade off between how soon the queen begins laying and developed comb.