The Bug Blatter Beast makes a good point, and provides a statistically more rigorous approach.
A raw average of ranking, especially with small N (sample size) is flawed, which is why I set the minimum to 5, not, say 2.
But, as Indiana points out, Average of ranking is at least easily understood by most people, and probably more fun and interesting to them because of that.

The thing is, that looking at the results for top N from 5 to 30, a base average actually serves pretty well.

As to the actual formula proposed, putting in the factor for weighting based on players rank seems a bit tricky.
Firstly as a subjective measure (though such factors are common necessities in statistics).
Secondly progression, worth, value, contribution, throughout the ranks is not uniform, and to get good benchmarks for the weighting would take a deal of research (especially for me, not being as au fait with the intricacies of the ATP 'pyramid' as with that on the WTA) to set usefully.
Ideally, I would love to look at every match by every player, the ranks of the players involved, the level it was played at, the round, the score etc
From that, adjudge each performance on a meaningful basis - add all that up to gague actual worth of every single ranking point earned, and show which countries players most regularly exceeded expectation on that scale as well as their actual ranking etc.
I don't have a) the access to that data or, b) the time to do that, though I wish I did, as I think it would be fun and fascinating to see a data-driven system where you could justify every nuance.

So, instead, I've used JonH's original suggestion of average ranking, allowing for users to vary the size of N as they choose.

I don't think that any one chart is going to tell the whole story, so, I've tried to view it from several different angles.
Therefore, I also added the three other ATP views, to show relative positions of the actual players, and illustrate the variable gap sizes.
I think this is very illustrative of strength to supplement the raw average:

Again, you choose N yourself, and the countries, and dates concerned. This is Top 10 for the latest rankings for France, Germany, Belgium & GB.

At some point you would expect, or hope, that, all other things being equal, GB would 'catch up' to, or close the gap with, other similar countries, but this is not the case generally.

There is also something to be said in 'Strongest Nation' as to resources. Belgiums performance with their comparatively meagre population is remarkable.
So, add to that the Population data, tab, and, for example, pound for pound, Belgium punches far harder than GB, throughout all of the scales. GB also lags behind it's peers (ESP, FRA, GER, ITA etc) even at the top end, and progressively the further down the scale to look at strength in depth.

Overall from the 4 charts, I hope you get a useful picture of what is going on, and, crucially, from one data set I can download each week to just update the ranks, and show variations over time.

If someone wants to derive a meaningful weighting factor, then I can do a version using RBBOT's rigorous formula.

So many interesting approaches, and ideas. If the data were all easier to come by, I'd love to build a proper model.

First of all , massive thanks and kudos to Alibahbah for taking a little idea and turning it into some fantastically rigorous statistical analysis. Much appreciated and I will play around with a lot.

My original intent was to amuse myself and create a talking point, which I guess has been accomplished. At one level, I wasnt actually trying to find the strongest nation, rather being quite simplistic, taking the top ten, as Indy suggests, gave a reasonable proxy to both top level accomplishments and depth, whilst being easy to work out (dividing by ten) given I was using pen and paper.

I think the thing this does is gives everyone the chance to review and assess and discuss, which a discussion board is for, I guess.

My second thought is to the power of the resource - if I was a journo, there is some rich data here that a paper or ATP or WTA could turn into something interesting. I have mentioned previously that I have a friend who is a sportswriter for the Guardian (albeit Football) but a journo like Kevin Mitchell (one of the best tennis writers IMO currently, for the Guardian of course) may well see an interesting article in some of this data. ABB, would you have a problem if I pointed my friend, David Conn, from the Guardian to this data and suggested his colleague take a look?

thanks again!

Updated with rankings data of 30/01/2017: https://public.tableau.com/views/20170130ATPWTACountryStrengthsv2_3/ATPCountryRankings?:embed=y&:display_count=yes

For pure averages of ranking, based on top 10 players per country:

ATP

• Italy move past Australia into 7th place - a big rise by Seppi, and steady improvement from the Italian #'s 5-10 all contributing.
• GBR stay resolutely in 13th place, but improve their average ranking from 211 to 206. Improvements by Norrie & Broady the major contributors.
  • Things look unlikely to change for GBR for the time being. They have a 26 point average ranking cushion to Brazil, in 14th place, but also are 32 average ranking places behind Belgium, who are currently in 12th.

WTA

• The exploits of Ash Barty & Destanee Aiava sees Australia leap 2 spots, to #14, and improving their average ranking by fully 20 points in the process.
• GBR sit at #15, nudged down one by the Aussie surge; though average ranking is fractionally improved from 216 to 215.

Within the ladies, although we GB rank fell slightly, overall we are a fair bit closer to several runners and riders around the 200 mark, so maybe a chance to move up on that front. Again, hitting an average of 200 would a a good benchmark, but good that we are moving slightly forwards on mens and womens. Overall been a reasonable start to the year for GB men and women despite Andy losing early (which of course has no impact on the rankings) 

Thanks Aliblahblah. Just looking more closely I hadnt spotted the large rise for the Australian's on the WTA, significant rise from start of season. My first reaction was home court advantage, but of course every year they have home court advantage so that wouldnt have changed, clearly they are making a strong move.

Interesting re Belarus and glad the model is showing how those lower movements can impact on things. Most of the GB players in the top 10 currently are playing Challenger events these days so I guess we are unlikely to see mass shifts, more a gradual movement as one or two players have good or bad weeks? Presumably for Belarus their 7-9 ranked players where in Futures and maybe one or two events in the local region came off together and impacted the overall ranking?

AliBlahBlah wrote:

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Updated: https://public.tableau.com/views/20170206ATPWTACountryStrengths/ATPCountryRankings?:embed=y&:display_count=yes

Not much change on the ATP.
France and Spain still 1 & 2 respectively.
3 & 4 USA & Argentina again switch places; a weekly occurence, it seems.
Only real movement in the top 30 nations is Belarus, who drop 3 places from 28 to 31. Interstingly, this is driven by big losses for their players ranked 7, 8 & 9 - the model reflects strength in depth.
GBR stay in thirteenth position for the fifth week in a row. Avg. rank slides one point from 206 to 207.

WTA
All very quiet. Top 13 places remain as previously; USA miles ahead of everyone in the top spot.
GBR stay in fifteenth place, as with the ATP, average ranking slips one point from 215 to 216.

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 Based on this week so far, GB Top 10 average will improve to 205, Ed Corrie contributing well to that. If he gets further, we will see a decent improvement on the week as a top 10 average