My parents are from Bristol, which has had a large population for a considerable period. According to the Wikipedia article, Bristol had a population of 66,000 in 1801, and around 12,000 in the sixteenth century. Other provincial cities such as Manchester were probably similar. Pushing the family tree exponentially backwards, these numbers set upper limits.
In a city, out-marriage from the extended family is likely so I am comfortable that there isn't complete coalescence of the family tree back to, say 1700. Since that's ten generations, and factoring in some coalescence (50%), let's say around 500 ancestors in the tree in the oldest cohort.
From 1700 back to 1550 is another six generations. In this period, many people lived in rural villages, with significant inbreeding. On the male side, this is shown by localisation of surnames and also by genetic analyses.
Probably my 500 ancestors in 1700 can be traced back to villages surrounding Bristol and Oldham (Lancs), where my father's side came from. However, 500 ancestors will certainly include many emigres to Bristol (or Oldham/Manchester) who had come from all over the place. Let's guess the 500 ancestors came from 200 different villages, so each village gets on average 2.5 people.
In the six generations from 1700 to 1550, each villager will have 64 ancestors of the oldest generation. Assume a village size of around 128. Suppose several of my 1700 ancestors came from any particular village: they certainly won't each have a different set of 64 ancestors in that village; in fact the overlap will be extreme (see the paper referenced in the previous post here). Let's say they will share the same 80 ancestors of the oldest generation in 1550. In fact, let's use the figure 80 anyway, taking account of villagers who had come to the city and entered my family tree subsequently (post 1700).
So in 1550, over 200 villages, there would be 200 * 80 = 16,000 ancestors.
Now, we need to move this up a bit, as some of my ancestors would have stayed in the cities, which even in the sixteenth and seventeenth centuries were a lot larger than villages and where inbreeding would be less. It's hard to estimate how many we're talking about here, as outside of London, most people - prior to the enclosures - lived off the land. The result, however, would be less inbreeding, so let's move the oldest cohort of ancestors up from 16,000 to 20,000, as a rough estimate.
Then the total number of ancesters alive in 1550 would be no more than 20,000 + 10,000 + 5,000 = 35,000. Say 30,000 allowing for more double-counting due to family tree branch coalescence. This compares with the figure of 458,752 computed from simple backwards doubling over 16 generations - see the first post on the subject here.
According to the human genome project, the total number of genes in the human genome is around 30,000. Some of these are not variant, being highly conserved because they do something essential. However, genes are long sequences of nucleotides and typically differ between people at a number of different sites. So you could argue that each of my 1550 ancestors probably contributed on average less than one gene (allele) to me!
Finally, a graph showing the population of England in the mediaeval period. Click on the picture to read the text.
Note: Based on the paper (which assumes random mating) referenced in the previous post, How many ancestors? (Part 2), there are 14 generations between an ancestral tree covering 1% of the population and one covering 99%. On this basis, everyone who lived in 1200 AD in England and who has descendants today is an ancestor of everyone of English ancestry living today. Or using another result from the paper, my family tree includes 80% of everyone living in England in 1200 (the remaining 20% did not leave any descendants alive today).
