What is the percentage of people living in England in 1500 AD whose lineage is still alive?
Here is a partial answer that also expands upon a previous comment:

Between 1900 and 2013, 20,000 English surnames had gone extinct according to the Daily Mail's summary of research on surnames. This is a partial answer because it only tells us how many names are gone and is from the wrong time period. 

To completely answer your question, I suspect you need to conduct primary research. I would suggest the following broad steps:


Gain access to the data needed to answer your question. British Census records only appear to available from 1841 to 1911 so you most likely need to sample old names (e.g., graveyard based studies).
Use the data to parameterize a statistical model such as a Galton-Watson branching process models.


I doubt the answer to your question can quickly be found unless somebody has already done it. Although, if done well, it could probably be a masters' thesis. However, before embarking on such a project, I would suggest doing a good literature search to make sure it has not been done before.

Note: If anybody finds the answer, please feel free to take content from my answer and include it yours.