There's some interesting history behind why much of the world has the option of non-earthed plugs but the UK does not. As explained above, most systems that offer both earthed and non-earthed plugs for Class I and Class II equipment respectively, are designed so that both types fit the same earthed sockets.
In Europe, for class II equipment with low current requirements, the standard choice is the CEE7/16 Europlug, which is specifically the flat 2-pin plug rated at 2.5A with 4mm diameter pins angled slightly inwards. This is designed to fit all types of socket with 4.0-4.8mm contacts on 19mm centres, regardless of whether the socket has an earth. Thus the same plug suits countries with CEE7/1, 7/3 (Schuko), 7/5 (French etc.), Swiss, Danish, Italian, IEC 60906 (Brazil, SA) and obsolete types too. There are also higher-rated non-earthed variants such as CEEE7/17, which allow for 10A or 16A loads but only interface to a smaller selection of sockets. All of these, ultimately, are descended from and semi-compatible with an early non-earthed plug by Siemens, designed at a time when earthing was not widespread.
Then in the USA and systems derived from the NEMA standards, there is the 2-prong NEMA 1 type, which fits both legacy NEMA 1 outlets, and the standard 120V NEMA 5-15 and 5-20 grounded outlets. As with the European system, the basic non-grounded plug fits almost every outlet that has the correct voltage, and is descended from an original design by Hubbell that was created before grounding was commonplace. Less widespread but fundamentally similar is the Australian / NZ plug which can also exist in 2- and 3-pin compatible versions.
The UK's BS1363 plug is a special case for which there has never been, and cannot be, a non-earthed 2-pin variety. This is because the mechanism in many sockets, and as envisaged in the original specs, required the earth pin to open the safety shutters. If we don't need an earth pin for earthing, we must still have a plastic replacement for it, called an ISOD (Insulated Shutter Opening Device.) Part of the reason for BS 1363 being so different to many other plugs is that it was one of the most recent designs, from 1947, that was not compatible with any of its predecessors. But what about those predecessors?
This is where the old UK system starts to look very daft. Before 1947, we had three current ratings of plug (2A, 5A, 15A) with round pins, each available in earthed (3-pin) and non-earthed (2-pin) versions. But, although the line and neutral pin dimensions were the same for both versions, they were made to different standards and had different spacings between the pins. Therefore a 15A 2-pin plug did not fit a 15A 3-pin socket, and likewise for 2A and 5A. Thus there were six completely incompatible types in use plus a few specials like the Wylex.
So, in 1947 we had a re-think to get rid of this confusing array of different plugs, and came up with the fused, earthed plug that is BS1363. By this time, earthing was widespread, but the vast number of small Class II electronic goods that we use today didn't yet exist. So it probably seemed that there was little point designing a 2-pin non-earthed version of the plug that would complicate the shutter arrangements. As things have turned out, countries that had a non-earthed variant all along have ended up with a neater solution for small electronic goods today, while we have our later, 'better' plug that forces us to the larger earthed variant footprint even when not needed.