The effective reproduction number gives us the average number of people an infected person will transmit a disease to. Though Rt doesn't include a time dimension, perhaps related would be the average time between becoming infected to transmitting the disease to others. Are there any concepts in epidemiology along the lines of "average time to transmit an infection"? If so, what are they?
Yes of course; it would not be possible to create epidemiological models without some measure of time.
The equivalent of what you ask for is usually expressed as the number of infection-causing contacts per unit time, or as a total rate of contacts times a probability of transmission. Typically R is derived from this number rather than the other way around, since R is given by (rate of infected)*(time infectious).
The term "serial interval" refers to the time between successive cases; there are a number of similar serial interval statistics.
The clinical onset serial interval is defined in terms of symptoms: the mean time between symptoms appearing in an infected person, and the symptoms appearing in the infectees. This can also be used to estimate the serial interval empirically.
The diagnostic serial interval is the mean time between diagnoses for the infector and infectees.
The interval between successive infections is usually called the serial interval. The theoretical value of the serial interval is estimated to be the sum of the mean latent period (time from infection to infectiousness) and half the infectious period; whether this value has its own name, I haven't been able to determine. The "average transmission interval" is the sum of the mean latent period and the full infectious period.
See "The Interval between Successive Cases of an Infectious Disease" for a more extensive description.
For a prevalent class of epidemiological models, compartmental models, the population is partitioned into groups, such as susceptible, infectious and immune (called "removed"). The transition rate between susceptible and infectious compartments (∂(S/N)/∂t) corresponds to the concept in the question. With:
βbeing the average number of contacts per person over a unit of time,
Sbeing the number of susceptible individuals,
Ibeing the number of infected individuals, and
Nbeing the total population,