Bottom Line: There is some evidence that the tendency for a few given names to dominate is also seen elsewhere in Europe, even if the top names are not John and Mary
##Evidence Point 1: Lowland Scotland 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.
##Evidence Point 2: Swiss Emigrants
I have extracted all the given names from the archive.org text of Lists of Swiss emigrants in the eighteenth century to the American colonies. There are only about 2500 names, many more men than women. A lot of cleaning up of alternative spellings was needed: there were at least 10 different spellings of Catherine, for example, and I had to allow for the use of the diminutive "li" at the end of some names (this is a typically Swiss thing, used even today about all sorts of people and objects).
It is not clear how representative of Swiss society these people were, given that emigration was actively discouraged and the commentary in the book suggests that many of the people had fallen on hard times or were regarded as in some way dissolute.
With those caveats in mind, here are the top 20 names for males and females from this source.
Heinrich 15.40 Anna 24.70
Jacob 12.30 Barbara 13.90
Hans 11.70 Elisabeth 13.10
Hans-Jacob 7.83 Verena 10.20
Felix 4.88 Margaretha 7.77
Johannes 4.73 Regula 6.17
Hans-Ulrich 4.21 Susanna 4.34
Caspar 4.21 Magdalena 3.20
Rudolf 4.14 Catherine 2.86
Hans-Heinrich 3.84 Ursula 1.94
Hans-Conrad 2.51 Maria 1.94
Ulrich 2.22 Anna-Barbara 1.94
Conrad 2.00 Cleophela 1.14
Hans-Rudolf 1.11 Babeli 1.14
Andreas 0.96 Dorothea 1.03
Heiri 0.89 Barbel 0.80
Rudi 0.74 Veronica 0.69
Salomon 0.59 Esther 0.69
Junghans 0.59 Angelica 0.34
Franz 0.59 Adelheit 0.34
Again, the top few names account for a large fraction of the total, especially when you allow for all the hyphenated names starting with Hans. If they are counted together with Hans, that name accounts for nearly 32% of all males in this source, and the top five male and female names both account for nearly 70% of their respective totals.
##Evidence Point 3: Mainly Germany
Finally, using a dataset of mainly German male immigrants to Pennsylvania in the 18th century, we see a dominance of Johan/Johannes. This was a very large dataset so I have shown the top 36 (35 if you take Johan and Johannes as one). The top 6 (five if Johan and Johannes are merged) account for nearly half the males in this source. However it should be noted that some of these names represent the merged totals of as many as 20 alternative spellings.
Johan 20.30
Johannes 8.70
Jacob 7.21
Georg 4.89
Peter 3.33
Michael 2.94
Heinrich 2.72
Hans 2.58
Christian 2.54
Philip 2.20
Hans-Georg 2.11
Andreas 1.94
Friederich 1.91
Christoph 1.69
Conrad 1.64
John 1.55
Nicholas 1.54
Martin 1.37
Mathias 1.29
Job 1.22
Adam 1.13
Henry 0.97
Daniel 0.95
Caspar 0.95
Wilhelm 0.88
Ludwig 0.86
Valentin 0.80
Hans-Jacob 0.80
Joseph 0.77
Abraham 0.73
Hans-Michael 0.70
Anthony 0.59
Ulrich 0.57
David 0.54
Leonhard 0.53
Balthasar 0.53
If anyone would like the original raw data or the Mathematica notebook I used for the analysis, please let me know in comments.