Well, since in OTL Manchester is clearly third after London and Birmingham (I'm basing it on
this), you want to boost Manchester and/or knock Birmingham. I'm going to restrict myself to postwar PODs, since 19c Manchester was basically tied with Birmingham for second and PODs from that era could easily go either way.
The simplest possibility is to produce a trend that would merge Manchester and Liverpool into a single metro area. They're only 50 km apart, whereas Amsterdam and Rotterdam, which are part of the same metro region, are 70 km apart. So just posit that both cities suburbanized toward each other, until their urban agglomerations met. At this scale, you might not even need a specific POD - just run things a bit differently in the last 50 years and there you have it. If you do want a POD, posit that suburbanization specifically followed railroads, with investment in rail lines and subsidies for commuters. The Liverpool and Manchester would work as a commuter line for both cities, so the places in the middle would be joint suburbs of both cities. Moreover, rail investment would encourage the growth of Warrington as a junction station, forming a continuous Manchester-Warrington-Liverpool conurbation.
Another possibility: early devolution to regions in England would strengthen the regional capitals. This would boost both Birmingham and Manchester, but the Northwest/Greater Lancashire is larger than the Greater West Midlands, so Manchester would have more of a regional economy to tax. If Manchester centralized various functions that in the US and Canada tend to be run by states or provinces, like hospitals and the major area university, then it would be much larger. Of course, a larger Manchester is a Manchester whose urban sprawl has an easier time meeting Liverpool's...
A third possibility: in OTL, Birmingham remained a rich city until the 1970s, while Manchester declined after WW1 as the cotton mills moved to lower-wage places like the US. (That's why today the West Midlands have more immigrants than other non-London UK regions - immigrants don't move to where there are no jobs). So you want to produce trends that flip that. A less liberal UK government - whether socialist or corporatist - would do it, since Birmingham was a city of small businesses and Manchester was a city of factory owners and a proletariat; Birmingham's decline is in part due to taxes postwar Labour put on it. If postwar Britain fully adopted the tripartite system of negotiations among big government, big business, and big labor that Japan and most of the Continent used, then this would screw Birmingham while protecting Manchester.