This has been quite an amazing journey. I hope that you've enjoyed the look inside the Ministry, and discovered just how the Motorways were Numbered - a series of compromises, arguments and then sneaking things in through the back door!
The policy approved as you've read was the "sector" system of numbering where the sectors are bounded by the single digit motorways- the "tree" advocates, where motorways were numbered according to the single-digit motorway that they connected to had failed. Or had they?
Certainly the present system is based upon the "sector" system - but there are one or two odd twists in the modern network that makes you wonder whether there were still some "tree" advocates in the numbering department. There are several examples where the "tree" system could be said to be used - most notably the M49 from the M4, the M271 and M275 from the M27, the M180 from the M18 and M621 from M62. Whilst none of them (with the possible exception of M49) actually break the "sector" rules, they are certainly numbered consistently with the "tree" system when there are much lower numbers that could be used. In the case of M271 and M275; M21, M22, M24, M28 and M29 are all available and unlikely to be used.
To add additional fun into working the whole thing out, the sectors don't necessarily follow the current lines of the single digit motorways. Instead, it follows the single digit motorways (except for M2) as far as they were definitely planned in 1961. Hence, the 2/3 zone boundary follows the M3 as far as junction 8, where it then continues on in a straight line until it hits the coast - meaning that any motorway built in Plymouth would have an M2x number...
There are also cases where motorways have been extended "backwards" into the wrong zone, adding to extra confusion - an example here being the M62, where the western side started life planned as M52.
We've also seen that three digit M numbers were initially not allowed by the system. It was only when local authorities got the motorway bug (such as those in the SELNEC area) were three digit numbers allowed - and some smaller motorways had their allocated numbers changed, such as M64 to M602 and M65 to M621. And ironically, most of the three digit numbers such as M621 do not fit into the original idea for them!
And just to round the whole thing off, where possible motorways have been given numbers similar to the nearest equivalent A road - which was an idea that was thrown out quite early on. The fact that the M45 is next to A45, M25 is next (on the southern side at least) to A25, M4 and A4 both head west from London, and the M6 and A6 both travel from Leicestershire to Carlisle means that people question why the other motorways are not numbered in a similar fashion - most notably M69 which is completely parallel to the old A46 route that it replaced. Of course, the reason for that one is the the M69 is within the motorway "6" zone, whereas the A46 is in the all-purpose "4" zone...
So, what of those most confusing of motorway numbers - the Ax(M) motorways? As we've seen some people at the Ministry felt the whole idea of Ax(M) motorways was almost demeaning, that it showed that they were second class, and not as important as the Mx motorways. A number of people today think the same way. So - what are they all about?
Where a motorway was merely a by-pass on a recognised continuous route, such as A.1, it would not be given a separate M number but, in order to make clear its motorway status and that the special motorway regulations applied to it, the letter M would be added in brackets to the existing route number e.g. A.1(M). This would avoid chopping and changing of numbers along such routes.
Hence, A1(M) simply means a bypass for a section of A1 that traffic allowed onto a motorway can follow. There is no rule as to whether the Ax route itself should be downgraded, though this is often the case. People scratch their heads about the A3(M) in Hampshire as the A3 still runs parallel though Waterlooville, but it breaks no rules - it's a bypass on a recognised continuous route.
The motorway network was planned from the start on the premise that trunk roads which had already been improved to a high standard in rural areas would not be abandoned. In other words, motorways would not be built as a network (like the 'Autobahnen' in Germany) in complete disregard to conditions on existing parallel roads. Thus, in designing a motorway numbering system there was always a need to cater for corridors where town bypass motorways (such as the various stretches of A1(M)) would be glued together by lengths of all-purpose dual carriageway trunk road.
There's also the case of short spurs from the motorway network that have been given Ax(M) numbers, such as A308(M), A46(M) and A666(M). These are logically described as a section of a bypass for a route - to access the A308 from the M4, for example, using the motorway spur, the average person would cope quite happily with following the route number A308, with the (M) tacked on the end to indicate a motorway.
And what of Scotland?
As we have seen, the Ministry in London only had juristriction over the motorway network in England and Wales. Scotland went its own way entirely, and followed one of the rejected ideas from the Ministry - that motorways should be numbered according to the A road that they replace. Therefore the M8 replaces the A8, the M9 replaces the A9, and the M80 replaces the A80. It also explains why there is no M7 - there isn't a motorway that replaces the A7.
Several of the problems with that system have become obvious. For example, the M73 is quite a distance from the A73. However, the M73 (in combination with the M74) does the same job - to allow traffic to travel between Abington and Cumbernauld, though it gives very poor access to the intermediate towns on the A73.
As well as that, we've seen an example of what happens when there is a change in the A numbered routes. When the route from Perth to Dundee was reumbered from A85 to A90, then the M85 was also renumbered to M90, causing confusion to the motorway network around Perth.
Effectively, therefore, all Mx motorways in Scotland can be thought of as Ax(M) motorways...
For the answer to that question, we must again look at why Ax(M) motorways exist in England and Wales. More specifically, as by-passes along continuous routes which would then be renumbered as Mx numbered motorways when the long-distance route was completed.
Firstly, the A823(M) was only ever a short bypass section of a more major route heading both east and west from the unfinished ends as can be seen on its feature page. It is certain that should the motorway have been completed, it would have gained its full Mx number - as happened with the A8(M).
The A74(M) is slightly different. It was indeed constructed as small stretches of by-passes. During the upgrade work, it was a very strange experience as one minute you were driving along the old A74, then for a couple of miles you travelled on brand new motorway, then suddenly you were back on the old road once more. It was then obvious that you were travelling along the A74, but that some sections were motorway. When completed, it was due to be renumbered M6 (as could be seen from the "No M6 construction traffic" signs in situ at the time), though the renumbering has never actually taken place.
And Northern Ireland?
In Northern Ireland, the NI administration had juristiction. They went for a totally separate system, where motorways are unrelated to the A routes, although the M1 starts in the same direction as the A1, though it veers off beyond Lisburn. It also uses the 'tree' system which was considered unsuitable for England and Wales.
Of the original plans only the M1 and M2 got built to any considerable length. The M1 was built first and given that designation since it was the first NI motorway. The M2 therefore followed later.
Now here's where things get interesting - several intended spurs from the two major motorways were to be numbered using the 'tree' system. Each spur would be given a two-digit number, where the first digit would be the same as the 'parent' motorway and the second digit increasing from 1 sequentially as you got further away from Belfast. Therefore the plans were:
M11 - Dublin-bound spur (scrapped)
M12 - Craigavon spur (built)
M21 - Airport spur (scrapped)
M22 - Castledawson-bound spur (built)
M23 - Derry-bound spur (scrapped)
The Republic of Ireland
In the Republic of Ireland, motorways are very different animals. Under Irish law, there is no such thing as a motorway order in the manner of the UK, so all motorways are legally defined with an N number (or as an all-purpose route). M numbers only appear on maps and signs, not in the legislation. A ministerial order can be applied to any road prohibiting certain classes of traffic, which creates a motorway. Therefore, the M4 is a section of N4 under motorway restrictions. The "old" N route is then reclassified as an R route.