Sure, they'll have a caudex by spring. They pretty much start off with a caudex (just think of pretty much any other type seedling in comparison). It's not just a fat stem. During the first few months the seedlings concentrate on developing the caudex with a few leaves, but increasing in size.
However, just as a palm tree seedling doesn't look like a palm tree until it starts developing a trunk, the caudex isn't obvious as such until the Adenium seedlings start developing one or more vertical stems. Once the stem has grown a bit, the plant takes on a more bottle shape. This is probably the stage you are asking about.
As Kadie pointed out, there is no simple answer to your question. Much depends on the amount of sun, the temperature, soil moisture, and available nutrients. Seedlings in my experience can handle a lot of direct sun, though maybe not FULL sun (meaning direct sun all day long). Give them some protection from hot afternoon sun.
Also, keep in mind direct sun tends to dry the soil faster, particularly the top inch or so. Adenium seedlings have a very slight root system, and require consistent moisture (NOT wetness) during their first several months to thrive. If the soil dries out too much, even briefly, it can put the young plant into a stage of semi-dormancy. It will still grow, but more slowly for a while. This is likely where the notion comes from that seedlings grow better as a group than as individual seedlings, which I have also noted to seem true. The collective "canopy" of leaves helps slow evaporation such that the soil stays rather evenly moist all the way to the surface.
Cold will do the same thing. It is not just daytime temperature that is important. Daytime temperatures here tend to be quite a nice temperature for Adenium growth through the winter, but chilly nights cause the semi-dormancy I mentioned and the growth of my seedlings slowed way down.
We have a good eight months or so before "spring", so your seedlings, if well grown, can easily be 5-6" tall or better with quite obvious caudices. Actually, they should be quite cute little "bottle" plants before cold weather sets in.
Hope this helps!
