You could try making a passage from the log through your primed hive with a one-way valve in the passage. Then all the foraging bees would end up in your hive. But, the rest would be unlikley to abandon the brood unless you give them a reason to leave - is there any way to smoke them from the other end?
no need for one way valve, actually works much better without.. make your brood and extension of their home
if you place a hive (with drawn comb if possible) above the hole (or too the side if you cant place it above.. but with large diameter pipe leading to it and connect it to the hole)
in the tree and block all the gaps between it and the hive with aluminium foil and thick fabric so they have to go through the hive and exit through the top they will move in relatively quick...
there's a pretty high chance in felling the tree the comb will have been buggered in their colony, it probably isnt at a good angle for brood laying or honey storage on both sides anymore and the queen will be happy to move up to nice vertically alligned frames.
Once the bees have claned and (re)drawn the frame she will begin to lay there, place qx below the brood box once you have a good brood pattern. from then on all hatched inside the tree will gradually come out to tend to new brood. provided you supply enough supers the forragers won't bother to store in the log as it's a hassle to go in that far when theres comb ready narere the entrance laving more forraging time.
once all brood in the log has hatched the last of the nurse bees keeping their temp will also come out with them to the new brood and you can move the hive a short distance from the log and remove any links to it, then after a day for stragglers to make their way in (and for old colony to ahve lost sent etc), break a bigger hole or poke stick in old hive and wiggle it around and the forragers should "rob it" of it's stores into the supers on your hive. then your free to chop tree up + collect wax if you even want it..
this takes a fair amount of time though. depends where tree is and if they need to chop it up and remove it soon.