It would be very difficult to isolate the impact of the additional scurvy "survivors" given that it would be swamped by the overall growth of urban populations in Britain in the same period (thanks to general improvements in public health and wealth). However, it's possible to look at the general points raised in the question.
It's probably a mistake to assume that the large navy ports provided the bulk of the service's seamen. A table of the men 'recruited' by the Impress Service during the American Revolutionary War shows that just over a third of those recruited were from London (at a time when London held almost 10% of the country's population) but the rest were drawn from towns right across the nation. In fact, the second largest number of sailors was provided by Dublin, which provided twice as many men as Liverpool (whose contribution was almost matched by Cork). In addition a significant proportion of the crews were drawn from "foreigners" recruited from the British colonies and overseas territories (and from other European powers).
Of course, not everyone on a navy ship was necessarily a sailor. Most of the warship's crew were there to man the guns (and provide the power for pumps, capstans and pulling on ropes) which didn't require skilled sailors, just able bodies. So these "landsmen" could be drawn from all walks of life.
If we look at the losses of sailors in 1755 (which is about the time that Lind published his treatise) and compare it to the losses in 1812 (by which time scurvy prevention was practised thoughtout the fleet), we see that there is a notable drop in deaths as a percentage of those in service. However, the actual number of deaths is higher due to the much larger number of men in service.
In service 29,268 106,179
Died 2,236 (7.6%) 3,397 (3.2%)
Discharged 1,227 (4.2%) 11,848 (10.8%)
Combined 3,463 (11.8%) 15,245 (14.3%)
It's also worth noting the far greater number of men that were medically discharged/invalided from the Navy in 1812.
The discharge of a far greater number of seamen as invalids before they died reduced the number of sick men who would die in service. The credit for a reduction of the death rate was claimed by the navy's medical profession. But they can also be credited with the higher proportion of seamen who were discharged as invalids.
So it would appear that while the chances of dying at sea were reduced in the later period, it would seem that's partly accounted for by removing the men from the navy before they could die.
Skilled sailors were in equal demand by the merchant navy too and men frequently swapped between the two (voluntarily or otherwise). The increasing domination of the seas by the Royal Navy, meant that the British merchant navy was also the world's dominant merchant fleet. If we pick the same period as above, it grew from employing 38,710 in 1755 to 165,030 in 1812. So even in the midst of a long war, the merchant fleet employed a considerably larger number of men than the navy.
The merchant fleet also kept increasing in size post-war while the navy shrank considerably (in 1820 the merchant fleet employed 174,514 against the navy's 23,985). In fact, the navy workforce shrank by 75% in just two years. I would imagine that this post-war discharge of seamen had a far greater effect on ports and coastal towns than the few thousand extra men who were saved from scurvy. The merchant fleet absorbed some of this number but the rest would have to compete on the job market with everyone else.
1. The foundations of British Maritime Ascendancy, R. Morriss, p236-238
2. Ibid, pg251
3. Ibid, pg251
4. Ibid, pg227