Let's look at the proposition that although Western countries' populations in general are aging, global fertility rates will rise from current low levels at some point. This would be due to the idea that reproductive rates in western societies are down not because people no longer want to have kids, but because the part of the life cycle when it is seen to be economically feasible to raise children has shifted from the traditional reproductive age (20s-30s) to (40s-onward) and that as reproductive technology advances, childbearing age is being and will still be redefined to wider periods of life. As a result, we could expect to see an increase in the number of children of "aged" parents.
Just based on anecdotal evidence here in the US, it is fair to say that the shift in expectations as to when the ideal time to have children might be has definitely shifted to the 30-40 year old age range. And fertility technology has advanced to the point where women in their late thirties and forties are having children. There are some issues with the fertility technology, however. One is the increased likelihood of multiple births in a pregnancy. I don't have hard data at my fingertips but once again anecdotally couples that go to fertility clinics have a meaningfully higher rate of twins and triplets than couples who get pregnant the old-fashioned way. That could be a deterrent to the use of fertility technology, and damp the potential increase in total fertility.
Also, it is clear that women bear greater risk to their own health when pregnant their late thirties and forties, and there is increased risk of birth defects in the child as well. This also could dampen the use of fertility technology.
As far as ideal family size and the economic feasibility of having children are concerned, I believe that in countries today with low fertility rates, women have strong disincentives to having more children. Reduced personal income due to taking time off from work to bear and care for a child, and reduced personal time for studies, play, or work factor into the child-bearing decision. My impression of modern women of childbearing age in Europe and Asia is that careers and leading an independent life take precedence over raising children. I believe this is somewhat borne out by recent data found in a paper titled New Empirical Evidence on The Low Fertility Trap Hypothesis by Wolfgang Lutz, Vegard Skirbekk and Maria Rita Testa. A key finding published in this research identified by Claus Vistesen(which I paraphrase here) regards "the very recent evolution of ideal personal family size...based on data from Southern Europe and France. As we can see the decline might look ever so slight but once we get the figure in percentages it shows that the decline in some countries is pretty hefty given the fact that we are only talking about 5 years. Moreover, note that Italy's personal ideal family size is fast approaching below replacement levels and seeing that this is a lagging factor of actual fertility decline the evidence of a traps seems particular sinister here. Also of course the drop in Spain's ideal family size of about 9% is something to watch."
To sum up, it is not out of the realm of possibility that fertility rates in aging societies will rebound. Whether this will occur or not is to a large degree a function of social attitudes toward child-bearing. The trends noted in the study referenced above do not bode well for such a rebound.