The official QI Twitter account (@qikipedia) today said the following:
If you sent a letter in 19th century London you could expect a reply within 2 hours
Is there any merit to this claim?
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I interpret the claim to not be about how responsive a particular recipient could be, but to be about how long it took to deliver a letter, and how often this service ran. i.e. if the recipient was attentive, and quick to respond, how long would it take for a response to be delivered?
The Dictionary of Victorian London includes a number of quotes from contemporary sources about the London Postal service.
One source from 1844 shows that there are seven deliveries daily "in town", and they were collected every two hours:
Morning by eight o'clock, for the second delivery. Morning by ten o'clock, for the third delivery. Morning by twelve o'clock, for the fourth delivery. Afternoon by two o'clock, for the fifth delivery. Afternoon by four o'clock, for the sixth delivery. Afternoon by six o'clock, for the seventh delivery.
A source from 1879 shows it varied by postal district:
London is divided into 8 postal districts, in which the number of deliveries varies from 12 to 6 daily, between 7.30 a.m. and 7.45 p.m.
Again, collections occurred every couple of hours:
Take care to post before ¼ to 8, 10, 12, and 2, 4, 6, 8, in one of the Iron Pillar Boxes (first erected 1855) on the kerb stones of the leading thoroughfares.
This doesn't show that the letters would be delivered quickly, just that they were collected and delivered frequently. There may still be a large "lag".
A source from 1879 shows:
the third delivery in [Eastern Central District] [...] is made at about 10 a.m., and includes the letters collected in London generally at 8.45 a.m [...]
This shows that a lag of a little over an hour is expected. It seemed to get better later in the day (as they got over the backlog of the overnight deliveries):
The next nine deliveries are made in every district hourly, and include all letters reaching the General Post Office or the district offices in time for each despatch.
In summary, while it depended on the year and the region, once could expect to receive mail every hour or two, with a lag of around an hour or so. This doesn't support the idea that a response might be typically expected within two hours, but it does suggest that such response times would be sometimes possible on a good day.
In the 1850s, the postal system was reformed. From that time until WWI, there were twelve deliveries per day in London:
In order to improve the London system, the separate corps of letter-carriers who delivered the General and District mail were consolidated, there were twelve hourly deliveries per day, and ten separate postal districts were created (Daunton, Royal Mail, 46).
Letters in London: Communication and correspondence in the nineteenth-century city
People did expect fast replies, but it often took longer than two hours:
Victorians expected that a letter might arrive two to three hours after posting, although to the dismay of many patrons, it often took eight to nine hours (still far speedier than mail service today).
Posting it: The Victorian Revolution in Letter Writing
From the Oxford English Dictionary, the origin of the expression "by return of post":
By return of post (F. par retour du courrier): orig. by return of the 'post' or courier who brought the dispatch (obs.); now, by the next mail in the opposite direction.
This certainly suggests that if an urgent reply was requested, one would mark the envelope as such and the messenger would wait for you to write the response immediately. In such cases, a 2 hour delay would depend on the distance (London was about 5 miles across in 1850, based on this map). As other answers explained, conventional mail deliveries happened all day long - obviously the point there was that one would want to minimize the delay for getting letters delivered from when they were written. That doesn't prove that everyone would respond immediately to every letter that came in - again, unless it was marked to indicate that an immediate response was required.
Urgent letters would be marked with "haste, post, haste" - this is the origin of the expression "post haste":
1590s, from a noun (1530s) meaning "great speed," usually said to be from "post haste" instruction formerly written on letters (attested from 1530s), from post (adv.) + haste (n.). The verb post "to ride or travel with great speed" is recorded from 1550s.
Such marking would usually invite a rapid delivery, and quick reply.