The M41 does partially exist, in two little sections. One is the bit that connects the A40(M) "Westway" in West London to the A40 and the A3220 near Shepherds Bush. The other is the Tring Bypass in Buckinghamshire, which is now part of the A41.
The M41 was proposed to be a big part of the "London Box" plan. This idea was very continental, as it proposed to criss-cross the capital with fabbo motorways. It was far too late to do anything like that, as the idea of bulldozing large areas of London seemed to rankle with the indigenous population. It was soon scrapped altogether, but not before the little west London section had been built. From here heading north west, according to LR75, it would cross the M25 just north of Watford. It would end at the A41 north of Tring.
The importance of the M41 is difficult to work out. As it would straddle the A41, previously a hideously busy route, one might think that it just had to be built. However, when one considers that the M40, to the west, and the M1 to the east; are there, and both head in more or less the same direction, then one would rightly say that it would be pure folly to even consider it. I have never driven on the London bit of the M41, but due to its location, I presume that it is an integral part of the city's road network; and consequently totally necessary.
I have seen the London bit from the A40(M) as I went past, when driving a friend into central London about 10 years ago. It is raised up on concrete piles and has huge overhead gantry signs that say such things as "The South" in a big way, which make you think "Great, now we're really moving". Then it just ends.
The Tring Bypass is certainly the most interesting thing about the M41. As I pointed out in the reply to Andy Martin's email, Tring must have had a switched-on local MP or councillor to get that built. I previously mentioned the fact that historically, the A41 was a notoriously busy and consequently dangerous road. Therefore, a little town such as Tring would breathe a collective sigh of relief when the bypass was opened, as they would then be able to enjoy the peace that bypasses provide. It did look great in atlases, before being absorbed into the A41. I have never driven on this bit either, and now I will never get the chance, as, if it has been downgraded to A road status, then it is no longer a motorway. And therefore no longer interesting.
Tring bypass now part of the A41. Central London bit is still the M41, for all of a mile.
History has shown that due to the presence of the extended M40 up to Birmingham, and the M1 heading in more or less the same direction, the M41 is not required. It is sad though that so many motorways that start off as good ideas get shelved. Of course, all the poor people who would have lost their houses might have mixed feelings about this. I myself think that the Tring section should have stayed a tiny bit of motorway in the middle of nowhere, and I would have driven up and down it a few times for a laugh. This just shows how starved of comedy I really am.