During the Napoleonic Wars, when there was a shortage of qualified sailors (the "knowledge workers" of their day), sailors from overseas were permitted to join the Royal Navy; this is touched upon in "Jack Tar". The shortage of sailors meant that press-ganging was used to bolster the complements of many ships. Consequently, non-British sailors were not turned away.
At Trafalgar, HMS Victory had aboard a crew of 828 men. Apart from 700 English, Irish, Scots and Welsh, 18 different nationalities were represented on the Victory at the Battle of Trafalgar.
The following websites bear reference to this diversity:
http://www.hms-victory.com/index.php?Itemid=82
http://news.bbc.co.uk/1/hi/magazine/4110478.stm
http://www.dailymail.co.uk/news/article-353800/Factfile-Battle-Trafalgar.html
Furthermore, a docu-drama has been made on this subject by Justin Hardy, and was broadcast on Channel 4 on Monday 28th June 2010 at 9pm.
http://www.channel4.com/programmes/bloody-foreigners/articles/justin-hardy-interview
If you have a UK ISP, you can see the docu-drama via 4oD
http://www.channel4.com/programmes/the-untold-battle-of-trafalgar/4od
Consequentially, the residents of the Royal Hospital Greenwich were not limited to persons from the UK.
Below is a list of in-Pensioners admitted between 1824 and 1841 who are believed to have come from overseas. The data has been sourced from Kevin Asplin's website, which can be navigated to via the Links page.