David Howarth says that Nelson originally suggested "Nelson confides that every man will do his duty" but someone else suggested England rather than Nelson. He then called over lieutenant Pasco (the flag officer) and gave him the message, followed by a warning that it must be done quickly as he had one other message to put up. It was at that point that Pasco suggested replacing "confides" purely to save time as it would have to be spelt out.
As Howarth points out the original signal is more personal and therefore better, implying that he personally knew the crews would not let him down. As an aside I can't remember where I read it, but there is a rumour that many non-English on board the RN's ships that day muttered something to the effect of "Yeah, because he knows that we will do our duty, its the English you have to constantly remind".
Since the "Nelson/England" suggestion occurred before Nelson had called Pasco over I would be surprised if the choice had anything to do with speed of transmission, unlike the Expects/Confides decision.
That said the system only had a vocabulary of about 3,000 words or signals, which sounds a lot until you realise that the OED has something like 171,000 full entries. So those 3,000 had to cover a lot of ground given that much of that number would be common signals.
So it is possible that there was only an entry for either Britain or England, and its not implausible that faced with a choice between them, England would have been chosen. However there's no evidence either way.
That said I don't find West India Man's suggestion about the code book to be very likely. It would be the flag officer's duty to know most of the common codes without resorting to written help. If he had to look up a common word or signal all the time the messages would take far too long to put up.