My primary concern is the fact that it takes longer (days to weeks?) to oxidize than boiled linseed oil.
Weeks at least, possibly months. Indoors, if the conditions aren't ideal, it could take a year or more before the overwhelming smell of linseed oil dissipates1.
If I put the bed on the frame before it fully oxidizes/cures, is there a risk of combustion?
The risk of spontaneous combustion from linseed oil (and other drying oils, as well as products made from them including varnish and products like "Danish oil") is from any bundled rags that are wet with them.
If the oil is applied to the wood and then wiped away to nothing, as it's supposed to be, the amount of oil that might come out of the wood into the bed is going to be very small indeed. Enough to cause minor oil staining at worst.
Since you asked, you don't need to be overly concerned about the 'chemicals' added to most commercial BLOs. I won't sway you if you've made your mind up, but this is of potential value to future readers.
The metallic drying agents added to most versions of BLO are no concern once the finish is on the wood and it has cured2. The primary health risk is to the worker .....just as it is from wood dust, see next point.... and you can easily protect yourself from any contact during application by simply wearing gloves. There's no need to wear a mask of any kind.
In this case what makes the decision not to use a finishing product because of the chemicals it contains a little asymmetrical is there are compounds that are of concern in the wood used. The awesome smell of red cedar is from chemical compounds. And those in red cedar are actually a known irritant.
1 Only possible to generalise here, being a natural product linseed oil varies and it can be surprising just how much variation there is (the colour range alone gives an indication of this).
2 Where they are OK even if used for surfaces that will contact food, which gives a fairly realistic idea of the actual risk they pose from incidental skin contact with wood finished with them.