Apologies I didn't see your tag - I think the comma invalidated it.
There was a very good, long blog post a few years ago when a type of sling was on the market called "bag slings" - these were terrible for positioning and did unfortunately cause the death of several babies before they were banned. She was concerned about it and did some informal experiments measuring babies' oxygen saturation levels while inside these slings.
It is only on the internet archive now unfortunately so it is very slow to load:
<a class="break-all" href="https://web.archive.org/web/20110419195027/www.babyslingsafety.blogspot.com" rel="nofollow" target="_blank">https://web.archive.org/web/20110419195027/www.babyslingsafety.blogspot.com/
I found it reassuring (I think we might be wired similarly 
) because she explained that the baby would wriggle and grunt as their oxygen saturation levels would drop, and even if they did drop, they went back to normal within a minute or so of being removed from the carrier. Here are the sling guidelines she came up with as a result of it (live, so doesn't load slowly):
https://www.slingbabies.co.nz/Site/Positioning/Correct_Positioning_by_M_liss_Stelzer.ashx
The 30 minute car seat guideline as said, is based on a very small pilot study and the babies in the study were up to 4 weeks old (not 6 weeks). Some of them were also low birth weight/premature. To be honest, I don't think it's a very helpful study to compare to a modern, lie flat car seat especially with a healthy full term six week old. The car seat they used was very old fashioned (I dated it to the early 00s; the study was done in 2013 IIRC) and it was also set up incorrectly for a newborn, the straps were placed much higher, where they should be for an older baby and they didn't include the newborn insert in the seat, which consisted of a head support and angle adjusting wedge. Since people do use older seats/second hand and don't always know how to set them up correctly or lose parts of it over time etc, you could argue that they were going for the worst case scenario but to be totally honest, I think they just didn't know how to set up the seat properly.
The premise of the study was that the "prem baby car seat challenge" as done in UK hospitals at that time was typically done with the car seat in the "rocker" position on a floor, a position which is usually more reclined than the seat is when it is installed in a car. Due to the angle of rear vehicle seats and the necessity for crash safety, infant car seats can sometimes be up to a 45 degree angle from flat, and a car with the engine running obviously is a moving, vibrating surface, which is not the same thing as a static floor. So they wanted to see if there was a test they could run which would be more realistic. The results were so alarming that they had to abandon several of the tests, which is why UK guidance then widely changed to saying babies should only be in car seats for a maximum of 30 minutes in the first 4 weeks - even though that's not what the study was looking to find out.
This is not actually current guidance; probably because the reasoning is not very good. The two hour guideline was what was given previously, and so when the 30 min thing came in they said 2 hours after 4 weeks old. I have never managed to work out where the 2 hour advice comes from or what evidence it's based on. My guess is that it's just a number somebody plucked out of the air at some point, and it sounded about right so was used. Recently, I'm not sure exactly when, specific time limits have been removed from UK guidance and we now have advice similar to the US which is about not using the car seat as a sleep location outside of the car. My educated guess is that this is probably to reduce confusion caused by the two sets of conflicting guidance now that people share more car seat safety information generally on the internet.
I have read a LOT of studies about positional asphyxiation in car seats and as far as I can tell there has never been a case where the baby had been in the seat for less than an hour. I don't mean to say that it's impossible for oxygen desaturation to start to occur before an hour, because clearly it is, as seen in the 30 minute study and the bag sling observations. But essentially, it seems to me that the whole picture is somewhat like this:
Small babies have heavy heads and underdeveloped neck muscles. They can be at risk of oxygen desaturation when their airway is in a position where it is "kinked" (if you put your own chin on your chest, you can experience this yourself.) Or if they are lying over something, for example there was an early co-sleeping cot which had a possible set up which caused a small lip about an inch tall between the cot and adult bed, and a baby died due to the fact they were lying on the lip and it was putting pressure on their neck.
Even newborns do have an instinct that, if their oxygen levels start dropping, they will try to wriggle and move around. When they are unrestricted on a flat surface such as a cot, this is usually sufficient and they are fine, they are even fairly good at dislodging obstacles such as a blanket which falls over their face for example - there are studies about this in the context of cosleeping and SIDS. You can see this yourself if you put something like a muslin gently on your baby's face - they startle and move their arms around near their head, which usually dislodges it. (Obviously, you still want to avoid this because that reflex may not be sufficient.) The problem is that if their back and neck position and head is not well supported, because they are in a poorly fitted car seat or sling, or possibly even in a well fitted car seat due to the angle of it, their instinctive behaviour won't be enough to open their airways and oxygen saturation may begin to drop.
It's true that you can't necessarily tell by looking at a baby what their oxygen saturation level is, especially if they are asleep. But also, it's normal for oxygen saturation to fluctuate a little bit, particularly during sleep. We don't want it to fluctuate too much or drop beyond a certain level, but a small reduction in oxygen saturation is not a life threatening event, it's more of a warning sign. My understanding is that the PA deaths which have occurred in car seats typically happen very slowly over a longer period of time, and you can avoid the risk by limiting your baby's stretches in the car seat so that if there are any issues with their oxygen saturation level, they do not get to such a low level that they cause harm. In the UK where people typically do not drive long distances with babies and may walk around with babies in car seats on travel systems, the time limit of 2 hours was given perhaps to give some context to this, whereas in the US where people typically drive a lot more, the advice is that the baby should only be sleeping in the seat if it's in a vehicle. But the actual behaviour is the same no matter which approach you follow - don't leave the baby in the seat for an extended period of time, especially if they are sleeping. DO always use it in the car, because aside from very young, premature babies, the risk of a car crash is much higher than the risk of oxygen desaturation. The risk of oxygen desaturation increases over time, particularly after about an hour, so regularly having the baby spend long periods in the seat does increase their risk for this, and should be avoided, but it's not a certain thing - I remember a parent at a baby group I went to said she let her baby sleep in his car seat overnight, every night, due to reflux. This worried me intensely, but the baby was fine. I would definitely not recommend it, but also, it's not like babies were dying constantly in car seats before these limits were recommended. It took a while for the research to catch up to what the risk actually was because these deaths are very rare.
For myself I have always kept 2 hours as a rule of thumb for an absolute max although I would say most of the time it was very easy to keep a stretch in the seat less than an hour and I would probably stick to this day to day. We only really went to the 2 hours when we were doing long drives anyway. Since 2 hours is also a good recommendation for adults to get out and stretch their legs, I don't think it really has an end limit although I wouldn't be SO strict with it with older children - e.g. if we did a 5 hour drive, we would probably plan one stop rather than two.
For slings, I'd recommend going to a sling library/consultant just because IME they are SO good at adjusting the fit properly. But in general stick to the guidance above and the other guideline is to keep their face clearly visible to you and they should be high (close enough to kiss is a great guideline for front carries) and they need adequate back support not to slump down. The link is helpful. Unfortunately there have been asphyxiation deaths in poorly adjusted slings, so definitely recommend getting advice on this if you are unsure.