Local numbering plans with 3, 4, 5 and 6-digits were all in widespread use into the 1980s. All of these were straight numbers, with no letters involved.
Up until at least the very early 1970s there were even some old UAX5 offices (Unit Automatic eXchange) in rural areas with 2-digit local numbers. The UAX5 is quite a fascinating system in its own right, and in fact I've just acquired a GPO manual for it dating from 1930, but that's another story!
The changeover to all-figure numbers occurred during the 1966 to 1969 period. British dials of the time differed from American in having the letter "O' on the digit zero. That resulted in the disadvantages of not being able to use any office name beginning with "O" and that it was impossible to come up with anything for the 66x prefixes (or n66 for that matter). On the plus side, it meant that a zero could be used as either the second or third digit of a prefix, and allowed certain combinations of names which would not be possible under the American system. For example, London had both MONarch (606) and MOOrgate (600) exchanges.
In London, most of the older offices had names which related to some geographical feature in the neighborhood, such as the name of a suburb, a road, a park, a river, or other well-known building or landmark. Toward the end though, exchange naming had to become more creative, and names which were of no particular significance were assigned. The poetical/literary series of office names is a good example, such as BYRon (297), DICkens (342), and WORdsworth (907).
At the end of the 3L-4N era, there were approximately 250 prefixes assigned to offices in the London director area. Another handful were used as 3-digit service codes, e.g. DIRectory (347), TIMe (846), and TELegrams (835). The GPO even used UMPire (867) for recorded cricket scores at one point -- You'd KNOW you were in England then!
Of the 250-ish assigned C.O. prefixes, about 100 remained unchanged during the conversion to all-figure numbering, including many (but by no means all) of the offices serving the "City" and "West End" -- the central financial and business districts of London. For example, CHAncery, FLEet Street, GERard, and MAYfair all just went to their numerical equivalents of 242, 353, 437, and 629 respectively.
The other 150-odd offices were assigned completely new prefixes with a limited grouping of codes within a district. In the part of north London where my family lived, there were offices named ENField (363), ENTerprise (368), KEAts (532), and LABurnham (522). ENField and ENTerprise kept their existing prefixes, while KEAts and LAburnham became 366 and 360 respectively, putting all the 36x prefixes into the district.
Similarly, BARnet (227) and HADley Green (423) became 449 and 440, putting them in the same 44x grouping as the nearby exchanges HIGhgate Wood (444) and HILside (445), both of which kept their original prefixes.
The new prefixes were assigned such that no new code conflicted with a3L prefix which was being taken out of service, and in fact there was a permissive dialing period. The hitherto unused prefix groups such as 55x provided scope for changes where a whole bunch of offices was to be renumbered. In fact under the 3L-4N system the following "groups" were completely empty: 55x, 59x, 65x, 66x, 67x, 69x, 85x, 95x, 96x, 97x, 98x, and 99x (excluding the 999 emergency number). A further eight such "groups" had only one prefix in use, e.g. 46x had only IMPerial (467) and 56x had only KNIghtsbridge (564).
The 3L service codes migrated to the 1 level, e.g. 192 for directory and 151 for fault reporting. (And in fact the former ENGineers fault-reporting number was later used as the new 364 prefix serving Winchmore Hill, a part of north London in the 36x grouping I mentioned above.)
I'm not too familiar with the exchanges in the other director cities, but I know they also changed some prefixes during the transition to all-figure numbering. Birmingham, for example, renumbered some offices so that north/south/east/west/central districts were determined by the first digit of the prefix. They had an easier time making new assignments outside of London though as fewer prefixes were in use. Birmingham had, I believe, less than 50 central-offices at the time of the changeover. I'm not sure of the figures of the changeover arrangements for Edinburgh, Glasgow, Liverpool, or Manchester.
I don't recall ever seeing that number used in London, but it's possible. Sometime after the change to straight 7D numbering, and certainly by the early/mid 1970s the GPO was using the 246 exchange for a range of recorded announcements in London (and some of the other director cities). The most commonly used numbers were of the form246-80x1, i.e. 246-8021, 246-8091, etc. I can't remember all the assignments offhand, but they included local London weather, national weather, theatre information, and various recorded information lines for tourists.
In a lot of advertising in the 1970s, the GPO would use a 246 number on the phone dial, in much the same way as the Bell ads used 555-2368.
I know that "WEather 6-1212" (936-1212) is still used in a few U.S. cities, including Washington D.C. (202), Philadelphia (215), and Milwaukee (414).
The days of the phone company providing recorded weather information on a local number have gone over here. You can access it on an 09 premium-rate number from various private enterprises these days, but given the easy access to weather information from various other sources, you'd have to be pretty desparate to pay the equivalent of a dollar per minute or more to do so.