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George H, you asked:
"I wonder how big were those almanac errors in 1801-03"

To test out the amount of work involved, I tried out my earlier suggestion.
I grabbed a Nautical Almanac for 1803 (found on Google Books, with a little
difficulty). Then I picked twenty lunar distances at random. I wrote a
little bit of code that randomly selected a month, day, and hour, and then I
opened the almanac to the lunar distance tables for that month and found the
nearest available, legible lunar distance. I compared those published lunar
distances against the modern true lunar distances (using my web site
calculator --note that most other online calculators do not permit
calculation for Greenwich Apparent Time which was the standard until 1834).
The mean absolute error in the distances for those twenty distances was 33
seconds of arc. The largest absolute error was 67 seconds of arc; the
smallest was 4 seconds.

A few examples:
May 9, 1803 0600 (which is 1800 by modern time-keeping convention and that's
the way my tables read), LD Moon-Spica according to the NA: 60.10.8
(dd.mm.ss), true LD according to modern data: 60.9.45, Absolute error: 23
arcseconds.

Feb 15, 1803 0900, LD Moon-Sun by the NA: 78.43.50, true LD: 78.44.7,
Absolute error: 17 arcseconds.

May 24, 1803 0600, LD Moon-Sun by the NA: 48.22.16, true LD: 48.21.34,
Absolute error: 42 arcseconds.

Aug 7, 1803 0300, LD Moon-Aldebaran by the NA: 54.49.23, true LD: 54.49.33,
Absolute error: 10 arcseconds.

[I hope I transferred these correctly from my scribbled notes].

Each comparison takes about two minutes, and from my tests the biggest time
waster is scrolling about in the PDF file for the Nautical Almanac. A sample
of twenty comparisons is good enough for my taste (and the limit of the time
I'm willing to spend), but naturally if you do more comparisons you'll have
a better result. And it might be useful to break these down by the "other
body" in the lunar distance. Most of the error comes from the uncertainty in
the Moon's position, but a few arcseconds arises from uncertainties in the
positions of the lunars stars and, probably to a lesser extent, uncertainty
in the position of the Sun.

And you wrote:
"and who or what had brought them to light by 1813?"

The introduction to each edition of the Nautical Almanac in that era
included some comments from the Astronomer Royal (Maskelyne for the 1803
almanac; note that the intro for the 1803 almanac is dated "Dec. 2, 1791").
These comments actually include some fairly detailed information on the
particular tables used to generate the ephemerides. Sometimes the comments
read more like an advertisement rather than a scientific description, but
it's a good start.

And:
"It's an interesting thought, that if those discovered Almanac errors were
large enough to call for significant corrections to Flinders' observed
longitudes, they were affecting every other lunar navigator, elsewhere in
the World, in exactly the same way."

Yes, and through 1815 or so, they were probably a larger source of error in
the longitude than observational error for many observers. Note that for the
needs of live navigation, this wasn't necessarily a serious issue in this
period. Confirming or checking your longitude to the nearest half a degree
was all that most navigators seemed to require c.1800. But mapping, as
Flinders was doing, had to be done to a higher level of accuracy. And of
course once higher quality mapping data becomes available, then that drives
the need for better live navigation results. And round and round it goes:
navigation accuracy chasing mapping accuracy chasing navigation accuracy in
a sort of feedback loop that continues even today.

 -FER
http://www.HistoricalAtlas.com/lunars



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