I also calculated your proposed indicator of suitability of HSR links (namely, geometric mean population divided by distance) for the UK's 10 largest conurbations by population (wikipedia). Here's the result:

         from         to     score miles

90 Manchester      Leeds 41654.487  44.0

88 Manchester  Liverpool 38969.011  34.7

86     London Birmingham 36850.567 118.0

84 Manchester  Sheffield 30957.756  38.7

82 Birmingham Nottingham 23908.977  51.6

80 Birmingham Manchester 23736.151  95.3

78      Leeds  Sheffield 23732.979  41.3

76     London Manchester 20703.905 208.0

74     London Nottingham 18349.031 128.0

72     London    Bristol 18100.438 118.0

70     London      Leeds 17975.517 196.0

68 Birmingham      Leeds 15422.111 120.0

66      Leeds  Liverpool 15259.233  72.5

64 Manchester Nottingham 15065.351  81.1

62 Nottingham  Sheffield 14986.543  43.6

60 Birmingham  Liverpool 13961.131  97.8

58     London  Sheffield 13873.797 166.0

56      Leeds Nottingham 13618.395  73.4

54 Birmingham  Sheffield 13456.476  89.9

52     London  Liverpool 12319.395 211.0

50      Leeds  Newcastle 11661.980  98.5

48 Birmingham    Bristol 11542.302  97.2

46 Manchester  Newcastle  9683.194 145.0                                   

44     London  Newcastle  9605.132 281.0

42  Liverpool  Sheffield  9142.400  79.1

40     London    Glasgow  7716.775 403.0

38 Manchester    Glasgow  7455.181 217.0

36 Birmingham  Newcastle  6848.996 207.0

34    Glasgow  Newcastle  6714.830 151.0

32  Liverpool Nottingham  6584.736 112.0

30 Manchester    Bristol  6277.328 177.0

28      Leeds    Glasgow  6016.128 220.0

26 Birmingham    Glasgow  5632.880 290.0

24  Newcastle  Sheffield  5402.064 139.0

22  Newcastle  Liverpool  4761.271 178.0

20  Newcastle Nottingham  4669.284 164.0

18    Glasgow  Liverpool  4458.924 219.0

16      Leeds    Bristol  4370.254 208.0

14 Nottingham    Bristol  4297.704 141.0

12  Liverpool    Bristol  3725.906 180.0

10    Glasgow  Sheffield  3379.604 256.0

8   Sheffield    Bristol  3301.136 180.0

6     Glasgow Nottingham  3128.789 282.0

4   Newcastle    Bristol  2329.008 299.0

2     Glasgow    Bristol  2162.716 371.0

As was discussed in the US HSR diary, a minimum travel time of 30 minutes corresponds to about 86 miles. A hypothetical Liverpool-Manchester-Leeds train barely has time to reach cruise speed before it has to slow down for the intermediate stop, so the first HSR link that makes sense is London-Birmingham.

We have met the enemy, and it is us — Pogo