You perhaps don't need to be as cautious as you might think selecting for each use here because you're not hafting significant striking tools, e.g. full-size axes, mauls, or sledges.
Obviously you don't want to make any handles that you're sure are going to fail but while there are clearly better and worse woods it must be acknowledged that when need demanded — e.g. isolated locations, snowed in during winter, on the frontier — even felling axes were sometimes handled using woods that most authorities and modern experts (or 'experts') say can't be used for such a purpose. Not shouldn't, can't. And yet those handles may have worked for their users for a very long time.
This relates to a point I made in a previous Answer a long time ago on a similar topic: forget the species, what's this piece like? In short, a nice straight-grained piece of a lesser species could be superior to a mediocre piece of a superior species.
Some food for thought on this point here.
There isn't one requirement for the handles of chisels, ideal characteristics depend on intended use — striking (heavy use of mallet or hammer) or non-striking (paring, light use of mallet), as well as the type of chisel blade that will be mounted (socketed or tanged, and if tanged what type of tang exactly).
Even with the fairly narrow form of Japanese bench chisels, which include a substantial steel hoop at the end of the handle1, the four or so species commonly used are not at all equal. Japanese chisels are almost universally struck with hammers and even with the reinforcement of the steel hoop the less-durable woods do split with enough use, but this is known going in and we can adopt the same philosophy to some extent...... especially if the wood is effectively free!
Socket chisels are a little different to most tanged types because the socket itself helps to prevent splitting. This is why the range of species used for socket chisels both in the past and today is wider than with traditional square-tanged chisels. As few as three species were used for the vast majority of tanged chisels in some markets2.
So, for paring chisels you could probably use almost anything and get away with it, even pine might do (aesthetics aside of course LOL). I've seen a refurbed chisel set up as a parer but also used lightly with a mallet which had a new handle of cherry or a close relative and it has held up perfectly well in the years since it was fitted. Cherry is inferior to almost every species in your list.
For bench chisels, I would personally try to favour hawthorn and dogwood if suitable pieces are available. Apple could work well too.
For any chisel that you intend to use hard, hornbeam, black locust and blackgum are ideal. Reputation suggests handles in any of these three will be virtually unbreakable3, especially the blackgum. If you prefer the looks of any of the other species though just hoop the handle and you should be fine.... the new handles could easily outlive you.
Of course there are mallets and mallets, but I've seen hammer heads fitted to hafts made from lesser woods, even mystery woods, that have held up to use. Since you have your pick though you don't have to deliberately choose something lesser and if you want you could select based on the same criteria as immediately below.
Hickory is the obvious first pick here, since it's considered the preeminent handle wood for striking implements (and no longer just in North America, it has supplanted the former go-to choice, ash, in the British Isles and I presume elsewhere). However it's by no means the only candidate and hornbeam, rowan and black locust can apparently work very well.
Since the hatchets are likely to see the most sustained hard use do be more careful in your selection of starting material. Qualities such as the flow of grain, rounout (if any), grain orientation relative to the striking axis, the spacing of growth rings, are all factors that affect strength.
A few notes on seasoning
Once you begin to dry you should continue to do so. If you aren't drying long lengths of branch as they do in stickmaking your cut blanks should definitely have their end grain sealed to minimise wastage.
Hands down the best end-grain sealer for the home user is melted wax. Doesn't matter what kind of wax, so any old candle stubs, canning wax, toilet-ring seal can be used. Don't be the least fussy if the wax is dusty or dirty, or has soot on it, none of that matters for this (and when melted a lot of crud will settle to the bottom anyway). Apply two coats with cooling in between and you should reduce crack formation during drying by 97% or more.
But if you harvest fresh (and colder conditions are the ideal time to harvest new material since the sap is not running) you do have the option to boil your handle blanks if you have a pot of suitable size for what you're making. Read a bit on this here, where I first learned of it. Over the years since I've done this myself with great success both in a pot on the stovetop as well as in water in the microwave.
I've also force-dried wood in the microwave, but it's not something I would recommend unless you have a pressing need of dry stock right now AND you don't value the starting material too much. It's smelly work (especially if you overheat4), takes longer than you'd think.... which tends to lead to impatience and overheated wood, and there is no guarantee going in how much stock you'll lose to checks and other cracks.
1 As do some traditional Western chisels, including the 'registered firmer chisel' type.
2 Ash, beech and boxwood in the British tradition.
3 For anyone on the other side of the pond European hornbeam is not equivalent to the American variety. While it is still a fine wood it's not a match in terms of hardness or durability.
4 Even without overheating you'll get a noticeable smell doing this. If you overheat and you see visible steam you'll get a boiled-wood smell, and not only is this unpleasant at the time it hangs around in the microwave for months. DAMHIK ^_^