About your source:
This clearly is no scientific article, so to gather any trustworthy you would need to examine the sources of it.
On the article two sources were linked:

1. A wikipedia article about cycles in earth's astronomical properties.
2. a scientific article that only let me read the abstract which of course contains no data or anything.

The main claim of both articles is that there is a 41000 year cycle in volcanic activity caused by ice ages created by axial tilt(which is not the same as precession which you initially claimed to be the cause).
After searching for anything regarding volcanic activity during the past 100000s of years(and finding nothing), I decided to search for the ice ages first and found this:

At least the temperature seem to line up on the 40000 year scale starting roughly 30000 years ago.
So if the theory holds the next colder age should be in roughly 10000 years.
What you can also see in the graph is that temperature and CO₂-levels seem to line up(except from the cooler regions every 41000 years). This also suggests that the 41000 year colds are created by external factors. But it also tells us that higher CO₂ levels lead to higher temperatures. Also the correlation of CO₂ and temperature seems to be very linear. So based on the graph I think you could say that a change in CO₂ of 100 ppm corresponds to a change in temperature of 8K(I don't take into account here the effects that temperature has on the amount of CO₂ being released. The actual value would be more something like 1K). Assuming a linear growth of CO₂-levels(of 1.5 ppm/a) we would rise our CO₂-levels by 150 ppm/100 years. So after the 10000 years we got left to the next ice age, that level will rise towards a much higher level that will be completely able to reverse the cooling effects and make them unmeasurable.

As you state jet streams lead to an actually colder climate in the USA(that's probably why most climate change deniers come from there). Also on most places in the world the warming is quite visible. Where I lived there used to be a 2 month long period of snow in winter, but during the last decade or so the amount of snow decreased(currently there are a few days of snow left). Also the summers here keep getting hotter each year(on average).