Quick note that Preswick is not really Glasgow (35 miles away) and Glasgow has its own airport which presumably was also affected by the same weather so they couldn't divert to that. Between the Scottish lowlands (where they had already tried all the commercial airports) and anywhere else, Manchester is about the closest option.
Carlisle is small (and not currently licensed for public use) - not an ideal place to drop a 737 if there's a choice. It's also not that far from Prestwick so may have had similar weather. Newcastle and Teesside are both on the East coast and likely to be affected by similar weather to Edinburgh given the storm coming in from the North East. The next closest will be Manchester or Leeds/Bradford, with Manchester being larger, closer to where passengers want to go (Glasgow) and further away from the storm.
There's precedent for this kind of situation to generate quite extensive investigations. An incident in 2017 where a flight from the Isle of Man to Belfast was unable to land in a storm, diverted back to the IOM, then landed in unsafe weather conditions because of insufficient fuel to divert again got a 48 page report[0], safety recommendations, and the airline being banned from the UK.
[0]: https://assets.publishing.service.gov.uk/media/5a82ede440f0b...
> Or was the weather also bad there?
That's likely, these places are not very far apart, and weather systems that cause 100mph winds don't tend to be small. And presumably if you have at most one landing attempt remaining you don't want to be taking any more chances.
As someone totally ignorant of British airports, a Google maps search for "airports northern england" shows Teesside, Carlisle, and Newcastle all significantly closer to Edinburgh than Manchester. Are these not places where a 737 under emergency could land? Or was the weather also bad there?