Coming from a country where an „integrated Takt“ (in local parlance, a regular Takt is simply a clockface timetable, it is the integration part that really makes the difference) is slowly being put into practice, just a few thoughts.
Out of curiosity, I have started to sketch what an integrated Takt for the UK would look like. Before you start criticizing me - I am perfectly aware that it is just a crayonista exercise and whatever I come up with has zero chance of working in reality. Just see it as my personal Sudoku...
However, some points: starting with the Southwest to Northeast Crosscountry is logical and probably unavoidable, since it connects all major routes from London as well as the Trans-Pennine routes. Places like Birmingham New Street, Sheffield, York etc are logical hubs. If it means that the London to Edinburgh trains leave KX at 17 or 43 past the hour instead of 0 and 30, it doesn’t really matter and will not change the usefulness or attractiveness of these services one bit.
The fact that most journey times between hubs in the UK are not multiples of 30 (journey times just under 30 minutes between hubs would be the perfect precondition for a nationwide Takt with at least a half-hourly interval on every route) is not a problem in itself either. It just means the Takt would not be perfect, but what actually is? Even the Swiss one is not perfect, and they have been building their infrastructure for Takt purposes for the last 30+ years. So you will find better solutions on some routes (eg the accelerated XC schedules Birmingham - Sheffield that run - or used to run pre Covid - a few times a day, which gives a 60 minute journey time between the two cities with Derby at half-Point) and imperfect ones on others... not a huge problem.
What I quickly ran into, however, was limited station and station approach capacity. I am not even starting with New Street... Sheffield is a good example: In a perfect Takt, XC would serve Sheffield exactly around minute 0 and 30, with TPE services built around it, London services arriving a few minutes before XC and local services arriving 10-15 minutes before the “hub“ and leaving 10-15 minutes after „the hour“. I don’t consider it feasible - even though my limited knowledge - to build such a timetable with two-track approaches on both sides and the limited number of platforms - and this is before you think of all the other factors everywhere else, such as London approaches, the Castlefield corridor (for TPE), all those flat junctions etc etc.
Trying to build a real integrated Takt in the UK would therefore most probably require station investments on a very huge scale and over a very long period. The UK network in this regard cannot be compared for instance to the German one (where they are slowly trying to integrate their regular interval timetables), where station and approach capacity are in most - of course not all - cases much higher.
Probably means that focussing on „only“ regular interval timetables (which is of course the reality on most major UK routes anyway, even though not to the same degree as the Swiss) and the most important connections (which is also done anyway) is probably the most realistic and useful way forward for the UK, especially as you focus on very frequent services (if north TPE runs every 15 minutes, perfect connections lose their usefulness as another train will come along shortly anyway).
As I said, just an outside point of view - I hope nobody feels I overstepped the mark.
As someone who lives in Sheffield, I think you make a lot of valid points about how to start planning a proper timetable.
The main XC route seems the best place to start - put that in the timetable (with appropriate dwells at major stations, but nothing like some of the awkward ones that XC have had, as they wait to slot into the next main line that they reach/cross), and then let each region arrange its long distance services around the XC ones that are "Set In Stone" - do the regional stuff next.
The problem that I have with all of the "takt" stuff is that people start from a very well meaning position but struggle to get everything to fit as neatly as they'd like in their heads. For example, you can arrange an infrequent branch line to have good connections at the junction (e.g. Par for Newquay). But that's not possible at busy stations. So you get a situation where people complaint that the timetable is poor because there's an awkwardly short or tediously long connection between the "London" train and the "local" train but this is often because the "London" train cannot connect with every single "local" train.
For example, at Leeds, there's a two track railway towards the north west towards Skipton with branches for Harrogate/ Ilkley/ Bradford. Given the limited capacity on that line, and the need to provide a balanced service, at least a couple of those destinations are going to end up having pretty poor connections with London services in at least one direction (e.g. the half hourly Ilkley service may arrive in Leeds two minutes before the half hourly London train departs, which would be an insufficient connection, so means a thirty two minute wait).
But you've got to base a timetable along the fact that many people just make unremarkable shortish journeys on a regular basis. And that means having a well balanced service from Leeds to Shipley (balancing the Skipton and Bradford trains apart) rather than having two trains within a couple of minutes of each other and then nothing for almost half an hour, because you've written the whole timetable around optimising London (or other long distance) connections.
Or you get the alternative, where people complain that the infrequent "branch" service doesn't connect with the "right" main line service - e.g. the arguments on here about whether the Windermere shuttle should connect with trains towards Manchester/ Birmingham/ London.
Connections are one thing where it's a stand-alone branch line (like Par - Newquay), but another thing where you have services meeting at a "cross". For example, you have an hourly Birmingham - Shrewsbury - Aberystwyth service and an hourly Cardiff - Shrewsbury - Manchester service. Should they connect to permit a Cardiff - Aberystwyth service or an Aberystwyth - Manchester service? Or do you make both services wait best part of ten minutes at Shrewsbury to inconvenience people staying on the same train, for the sake of people trying to change trains in every direction at once? (e.g. the suggestion on here recently that services have a long wait at Salford to permit people from north/west of Salford to have connections to both major Manchester stations - great if you want flexibility but do passengers from Bolton to Piccadilly really want to wait in Salford for over five minutes to ensure that connections work in both directions? Should Sheffield - Lincolns services stop for ten minutes at Retford to provide a good connection in both directions with ECML services?
And then there are rural routes where you'd be a lot better forgetting about high minded concepts and focus more on having a good timed arrival in the "town" station. Much more important than a bi-hourly service.
Plus the complicated pattern of hourly services that we have because we insist on direct trains. e.g. in Sheffield you have two "semi-fast" services per hour to Manchester/ Nottingham/ Doncaster/ Leeds - if that was a half hourly Manchester - Doncaster service and a half hourly Nottingham - Leeds service then fair enough (although you might have to argue about whether to prioritise a particular connection in one direction, or occupy four of the five through platforms at Sheffield simultaneously so that you could try to arrange connections). But you can't keep all of the through services at the same time as having a well balanced half hourly service in each direction - what's the priority?
Good on paper but hard in reality.