You could consider another option: don't refinish it at all. Depending on how damaged the existing finish it (you don't say or show) in most case light scuffs and dings are just part of the charm, and don't represent any problem with ability of the material to resist moisture.
That veneer is very stable, with very good natural water resistance. On top of that, it was probably finished with some sort of decent outdoor spray polyurethane. So, unless it has been treated with complete disregard, it'll be fine as-is.
You wouldn't want to leave this out in the elements permanently or seasonally, but the occasional ride through a sprinkler or being caught in the rain isn't going to harm it. But the assumption is that most of the time this bike is brought into a porch or garage.
Outdoor furniture has to be refinished regularly because it is left to the tender mercies of 2-3 seasons of weather, and hundreds of hours of UV light exposure. Unless this bike has had the same treatment, it's probably fine.
That all being said, if you want to refinish these pieces with most finishes, including the penetrating finish you mention, you will have to remove all of the previous finish.
- Take everything apart.
- Mask or remove the hardware. Some of those bearing surfaces might be screwed in, or be a really tight interference fit. However you do it, you want to get these as clean as possible and maybe even remove the rust.
- Using an appropriate remover, remove all the old finish completely.
- Repair the damage. This is not going to be easy, but you might have to get clever with epoxy and canoacrylate glue and clamping.
- Hand-sand to a smooth finish. The idea is to sand for comfort of the person using it, at least with the finish you have indicated you want to use. Penetrating finishes don't really require any surface prep, other than it being bare wood.
- Apply, re-apply the finish as directed.
Of course, all those decals will be removed, and the hardware and fit might not be very tight. In some cases you might even have to rebuild up some wood "bearing" surfaces with epoxy or a wood insert that is machined back to nominal size. As mentioned before, you are going to have to make sure the metal bearing surfaces don't spin where held in wood, and are nominally the right size to meet each-other.
A note on finishes:
There are pros and cons to using a penetrant finish vs. a sealing finish, like a harder exterior polyurethane (or varnish). The former is intended to be re-applied regularly, and is also intended to age and patina over time. A sealing finish will need to be refinished less often, and is intended to cure hard on the surface. Penetrating finishes need to be applied to clean, absolutely bare wood. Polys (and varnish type finishes) are slightly less fussy in this regard.
But the majority of the attention you will be spending on this project is removing the old finish, prepping the surface, and making sure the mechanical interfaces are nominal after years of use, exposure to the elements, finish removal, and surface prep. Try to stay away from the edges.
As for toxicity, pretty much any finish, once cured, is essentially non-toxic and completely safe. With penetrant finishes you have to make sure it isn't the same stuff that they use in pressure treated lumber, but they usually say that on the label. Those tend to be oil solvent based, as well.