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 expected within two hours, but it does suggest that such response times would be feasible on a good day.

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.