Chemical reactions in plasma state
Yes, chemical reactions do take place in high-temperature plasmas. In fact, these reactions present some problems in the analytical technique known as Inductively Coupled Plasma - Mass Spectroscopy (ICP-MS), which typically uses an argon plasma.  

As noted in the comments, the conditions within a plasma are very different from those in a laboratory at STP, for example. But that just means that the chemical reactions that are observed are different, not that they don't exist. In ICP-MS for example, one is trying to detect the masses of very low concentrations of various elements in the presence of relatively huge quantities of argon, oxygen and hydrogen (the latter assuming that an aqueous sample is being analyzed). Species that are formed within the plasma include $\ce{Ar2+}$, $\ce{ArO+}$ and more. When one of these species has the same mass as the trace isotope of interest, it only takes a very small percentage of these products to interfere with the analysis.  

This example of argon plasma can largely be extended to your specific question "If hydrogen and oxygen plasma were combined, would the separation of electrons from nuclei prevent normal chemical reactions?". Depending on the conditions of the plasma (temperature, concentrations, degree of ionization, etc.) the answer is that the reaction would only form some small degree of transient water molecules, as compared to say molecular hydrogen and oxygen gas at high temperatures or in the presence of a catalyst, which would overwhelmingly form water. In other words, the former (plasma state oxygen and hydrogen) would form trace amounts of water, and probably hydrogen peroxide and other oxygen-hydrogen species, where under "normal" conditions the formation of water would be strongly favored (thermodynamically anyway). It's just the nature and definition of a plasma that whatever species you have in your plasma will want to be broken into it's elemental constituents and the formation of molecules will be strongly unfavored.

Summary:
The conditions within a plasma are very different from conditions typically encountered in a non-plasma environment, and thus the chemical reactions are different. The formation of otherwise exotic species like $\ce{Ar2+}$ are relatively commonplace within an argon plasma for example ("common" being a relative term of course, as concentrations of $\ce{Ar2+}$ may still be relatively low, just much higher than would be seen outside of the conditions present in the plasma). Plasma conditions themselves can also vary widely, from conditions where only a small portion of the gas molecules are charged to conditions where the majority are charged. Regardless, the high energy state of a plasma tends to result in the destruction of molecules to their atomic precursors, but occasionally there are reactions to form products that would not be seen outside of a plasma.