Thomas Philip Adams might be British after all, and even fits into your Trafalgar/India timeline.

There is a record of him being a licensed school teacher in The Hague, aged 45 in 1813.
This would put his year of birth 1767/8. There is a death record of Thomas Philip Adams, dated 5 May 1831.

Father: John Adams
Mother: Magdalena Morrin
Deceased (male): Thomas Philip Adams, 63 years old
Spouse: Frances Sarah Allkin

Source: Nationaal Archief / Rijksarchief Zuid-Holland te Den Haag, BS Overlijden
Den Haag, 7 mei 1831, aktenummer 549

The teacher license looks like he was pursuing a new career, as others who received the same license were either half his age, or had prior lower grade licenses. The names of the family members listed in the death record are not Dutch. His wife's name seems Welsh. Perhaps he had settled down in The Hague after his adventures.

Only question is: how would a Prussian have known about his exploits in November/December 1813? I have seen Adams' name being mentioned in a Dordrecht newspaper, so maybe his actions were seen as something special.