Which country allows IAPs without ATC?

In the US, at least, there are a lot of class E^ airports without towers, and it appears to work fairly well on the basis of voluntary deconfliction via CTAF.* In VMC the inbound IFR flight declares its intentions and asks people to play nicely by spreading out their patterns by going upwind longer or whatever, which usually works. Otherwise, the IFR flight has to either go missed very early, or switch to a VFR flight and join the circuit or do a straight in landing, depending on if they are training or actually trying to land. In IMC I don’t think it’s much of an issue, because IMC requires IFR, so all flights are separated and controlled at any place with an instrument approach.

*Commercial flights generally have more control (B/C/D airspace) at their endpoints. There are a few exceptions, like a casino 737 flight into KENV, but in those few cases the commercial traffic has to play by the same rules as a Mooney as far as a 172 in the pattern. To be sure, most pilots are fairly willing to play ball, so in practice it doesn’t seem to be that much of an issue. I suppose there are edge cases of somebody pottering around NORDO in MVFR, but in practice it seems to work out okay.

^ I.e. airports where traffic deconfliction is entirely on the pilots, and more relevantly airspace where ATC has no authority over VFR traffic.

In the US, most airports with instrument approaches do not have towers. The vast majority of the non towered airports have class G airspace to 700 AGL A minority of non towered airports have class E to the surface.At class G non towered airports, VFR is allowed as long as the visibility is 1 SM and clear of clouds, so if the conditions are 300 overcast and 1 SM, it is a beautiful VFR day in the pattern. On the other hand, if the airlines fly into the airport, it is likely that the non-towered airport is class E to the surface. Class E VFR flight at a non towered airport requires 3 SM visibility and a minimum ceiling of 1000 feet and standard cloud clearances of 500 feet below and 2000 feet laterally, So anything lower than VMC, there will only be IFR traffic in the pattern. Operation at non towered airports in class G and E do not require a radio. There are instances of towers in both Class G and E airports, but G is usually temporary towers for events. It will take a mid air or two to change the rules for class G airspace, one if it is a VIP. But so far there haven’t been any where an IFR aircraft breaks out and finds a VFR aircraft in their windshield.

Peter wrote:

AIUI the UK concern was with IFR traffic popping out of the cloud 100ft above the pattern (circuit) height, which doesn’t give the pattern traffic much time to spot it

In many European countries including Germany, this threat is supposed to be mitigated by the RMZ’s that surround every uncontrolled field with an IAP. Of course this works only if everybody makes correct, useful and timely position reports and acts upon those of the others.
As said in post #21, in the US, someone could legally be NORDO in the pattern with an airliner on final approach – even if this is not an issue most of the time, I like the European solution much more in this case. Whether you need an AFIS person for that, is an entirely different debate…

Peter wrote:

That’s an elegant solution which, together with a State funded approach controller, evidently does the job pretty well

How does this even address the issue of someone being in the pattern in MVFR and another plane coming out of the clouds at TPA? Class E does not require radio contact, it just raises the VFR minima a bit.

Peter wrote:

while the UK CAA has or is proposing to cripple that option with

In CAS (class A-E) below 3000 MSL or 1000 AGL SERA requires 5km vis, 1500 m horizontal and 1000 ft vertical cloud spacing. Outside CAS (class F/G) at those altitudes, 5 km and clear of clouds is sufficient, the national CAA can specify a lower vis (down to 1500 m) for flights at 140 KIAS or less.

Before SERA the RMZ’s in Germany used to be Airspace F, which had, despite being uncontrolled, CAS VFR minima. This is not an option any more under SERA (anything OCAS below 3000 MSL / 1000 AGL is just “clear of clouds” today), but in my opinion cloud clearance requirements are by far second best to mandating radio calls when it comes to IFR/VFR-separation at uncontrolled airports.

In practice in the UK all airports in CDE (of which only D is used) have towers, so I don’t see why allowing for less stringent VFR minimal is crippling in any sort of way.

Peter wrote:

Did you perhaps mean 3000 feet?

No, I meant 300 feet. As long as the visibility is 1 SM, and you remain clear of clouds, it is VFR. The point is that without a cloud clearance requirement, you can in theory have a legal mix of VFR and IFR aircraft anytime the visibility is at least 1 SM.