Considering that modern economies rarely experience recessions due to supply shocks (such as spiking oil prices), we should look at this from the perspective of a demand shock, a decrease in spending. This should not have too significant of an impact on the supply of exports. The supply for these depends on foreign demand, theoretically unaffected by the recession in this country thus far. Some firms may produce both for export and for domestic demand, and some of these might need to shut down if prices drop below a certain level, but this should not occur on a large scale. The fall in domestic demand may even be largely compensated for by foreign demand, with a slight fall in price. The "high wages" would be no different from before the recession, and they would eventually fall as the economy comes back to potential output.
In contrast, net imports will definitely fall if aggregate demand does. Considering that spending on these is tied to, well, the same factors influencing aggregate demand, anything which might have induced a contraction of spending on domestic goods would similarly decrease demand for imports.
Given an at worst moderately impacted supply of exports and a massively impacted demand for imports, the net export of the recessionary country would certainly fall (though this may be more ambiguous if there is expansionary monetary policy to counteract the recession, but this might work in the same direction). The major trade partner would, of course, see its net export move in the opposite direction, importing at similar levels and exporting less. The key thing here is that the target country's demand for imports would not be modified.
