I would have thought that broadcasting live from St Paul's at short notice would be quite a technical logistical challenge - were there any already planned broadcasts from there over the Bank Holiday weekend?
I'm not sure - but St Pauls is a regular venue for broadcasts, and the BBC - at that time - had an in-house Outside Broadcast resources department (*) and would have been very familiar with how to set-up and rig for a broadcast at that venue. There will almost certainly have been a camera plan from a previous occasion that could be used as a model - and modified if need be.
Impressive - and hard work - but not unheard of. The BBC were - and are - quite good at this kind of thing.
(*) And to be fair even though the BBC no longer has an in-house OB dept, they still have a talented team of in-house and freelancers, and the third party resource providers they now work with are excellent.
St Pauls never had permanent connectivity, it would all have been microwave links into either Crystal Palace or the Swains Lane receiver. How do you do these things at short notice? Well, organisation and a decent filing system. Something that nowadays is far less common, jobs are changing while we are rigging them these days.
For the links, we had a large map on the wall with numbered pins in it showing sites we had used previously, and the midpoints they worked into. There was a file for each site, detailing contact names and all the old planning sheets.
St Pauls would have had its own file, so we knew how to do the links, and the names of those who had last worked there were on the old sheets. So the riggers, Engineering Manager, Sound, Vision and Comms all had working knowledge and diagrams (sometimes quite old ones). You'd try to get those staff on the job as first choice.
To get links out was quite quick, load a van and with the agreement of the building owner get the kit on the roof (in the case of St Pauls). Cabling might take a while, plus parking issues may not help, but a national event can make the logistics move faster.
The internal cabling could be labour intensive, but enough people thrown at it and anything can be achieved.,