What changes in climate pattern can be observed when obliquity is at
While obliquity is easily googled, the specific effects globally aren't as easy to look up, so I'll give this one a shot.
For starters, the 41,000 year variation isn't precise. It fluctuates some, having to do with the earth and other planets not being perfect spheres and not all orbiting in the same plane of elliptic, and the planets (Primarily Jupiter), tugging on each other. The Moon stabilizes the Earth's obliquity somewhat, though that's often overstated. More recent models have shown that the Earth's obliquity would be reasonably stable even without the moon and in the future, the Moon could fall into resonance that will destabilize the Earth's obliquity. (I read that researching another question, but don't have the article handy and that doesn't really apply to your question anyway).
The easy part to look up is that Obliquity is primarily a polar and seasonal effect. Higher Obliquity (24.5% currently and the more extreme 30% proposed in your question) means colder winters and hotter summers in higher latitudes. (higher latitude means both poles, just for clarity, high latitude = far from equator)
High obliquity means more extreme seasonality (hotter summers and colder winters in both hemispheres). This does not change the year-average energy reaching the Earth, but in terms of temperature the Northern hemisphere is more important because it contains more land: Land warms up faster than water and it also hosts the reflective ice-sheets that contribute to Earth's cooling. As a result, Ice ages (somewhat) coincide with cold Northern Summers.
So, the simplified answer is higher obliquity, hotter earth, lower obliquity, colder earth.
Source - Milankovich
The tilt is in the decreasing phase of its cycle, and will reach its
minimum value around the year 11,800 CE ; the last maximum was reached
in 8,700 BCE. This trend in forcing, by itself, tends to make winters
warmer and summers colder (i.e. milder seasons), as well as cause an
overall cooling trend.
and Source - NOAA Article
less "tilt" means less severe seasons - cooler summers and milder
winters. The earth wobbles in space so that its tilt changes between
about 22 and 25 degrees on a cycle of about 41,000 years. It is the
cool summers which are thought to allow snow and ice to last from year
to year in high latitudes, eventually building up into massive ice
How extreme can the climate get if the obliquity oscillates between 20
and 30 degrees?
With a 20 to 30 degree variation in tilt, figuring the same 41,000 year cycle, Ice would build up in the North as the cycle leaned towards 20 degrees and ice would melt in the North as the cycle leaned towards 30 degrees. The "milder winter" that we might expect from a 20 degree tilt would be highly overcompensated for by the cooling effect of an ice sheet if there was an ice sheet, so, "milder at lower tilt", shouldn't be taken as strictly true but only true if there isn't an ice age coinciding with the lower tilt.
It's possible a 30 degree tilt, which would cycle upwards, 28 29, 30, 29, 28), that period of high tilt, lasting 10,000 years might be enough to melt Antarctica too, which for the most part hasn't melted in 15-25 million years or so. Needless to say, with a tilt that high, we'd probably go through more extreme variation with the rise and fall of ice ages and rapidly changing sea level, and ice age cycles are already quite extreme in terms of sea level rise, flooding from glacial melt melt and other changes. Anyone who was alive 12,000 - 8,000 years ago could verify that the changes were intense. Fossil records show an increase in extinctions during periods of rapid climate change.
The last maximum tilt happened 8,700 BCE, so the moving towards the maximum coinciding quite well with the end of the last ice age and the minimum is going to happen about 11,800 CE (or 11,800 AD for anyone who went to school when I did), and that (might) coincide with the rising of the next ice age, but it's a bit more complicated than that, as we've figured out how to stop ice ages by burning lots of fossil fuels, and the other orbital factors would need to be considered as well.
That's the simplified answer anyway. A more detailed answer would take into account the other orbital cycles (see Milankovich link above), the 100,000 year problem and the large permanent glaciers which survive the entire 41,000 year cycle, such as we currently have on Greenland and Antarctica and which affect albedo. I might try to tie that in later, but that would get longer and at my level of understanding, a bit less certain.