Lowland Scotland is not that far removed from England anyway, but I now have evidence that the same concentration exists there too. I have (laboriously) transcribed the Parish Lists of Wigtownshire and Minigaff, 1684, which amounts to more than 9000 names. The same tendency for a few names to dominate applies, but for women the names that dominate are different than in England. Here are the top-twenty male and female names, after regularising spellings. The domination of Janet (here together with its variants Jonet, Jannet and Jennet) is stark. Characteristic Scottish names like Fergus, Ochtrie, Niven, Grizell/Grissel and Elspeth are reasonably popular, and historic Scottish names like James, Alexander and Agnes even more so, but like in England, John and William are in the top two.
John 27.90 Janet 31.60
Alexander 12.40 Margaret 18.60
William 11.00 Agnes 8.81
James 8.27 Jean 6.30
Patrick 7.01 Marion 5.59
Andrew 6.24 Elizabeth 5.04
Thomas 4.46 Isabel 3.49
Robert 3.90 Grizell 3.21
Gilbert 3.00 Helen 3.00
Hugh 2.10 Catherine 2.68
George 1.92 Christian 1.64
Archibald 1.51 Elspeth 1.49
David 1.31 Mary 1.43
Anthony 0.77 Ellen 0.96
Michael 0.70 Sarah 0.85
Fergus 0.59 Bessie 0.85
Adam 0.54 Barbara 0.57
Ochtrie 0.36 Marie 0.53
Niven 0.32 Anna 0.51
Ninian 0.27 Katrine 0.43
I note that the less common but characteristically Scottish names like Katrine, Niven, Ochtrie etc tended to come in runs in the parish lists, meaning that they were more like to be in the same village / family / household. Even if they didn't have the same surname, they might have included sons of daughters of a man with the same name (and analogously for female names), and/or people named after a prominent person of the previous generation, who was not related.
Statistical testing wasn't consistent with a true power law in these frequency data, but it was reasonably close to one.
If anyone would like the original raw data or the Mathematica notebook I used for the analysis, please let me know in comments.