Basically relies directly about which plant-legume interaction you are interested. In common bean and soybean the interaction between rhizobia and root plant legume give rise to changes in the root that trigger the nodulation process and it is possible to view young nodules since 10 days post-inoculation (or even to determine the presence of nodule primordial since 7-8 days post-inoculation).
Depends on the rhizobium and plant species you are going to inoculate. For redgram, black gram, groundnut etc., it takes a minimum of 20 days to visualize the nodules.
When tested under gnotobiotic culture conditions, internal nodule primordia (foci of cortical cell divisions near sites of primary infection) can be detected within white clover roots inoculated with the appropriate rhizobia and cleared using the Truchet technique (clearing with diluted chlorox under vacuum followed by methylene blue staining) in as early as 4 days after inoculation. Nodules typically begin to emerge at day 8 or 9 and begin to fix dinitrogen around day 12. Details will vary based on the legume-rhizobia combination.
It depend indeed on the type legume species. When looking for the host plant range of some rhizobial isolates, I remarked that some nodulated in few weeks whereas others took some mounths to nodulate, using the same isolate...
The best example is that of Retama sphaerocarpa (in another study) which take more than four mounths to show small nodules. However, the bradyrhizobia associated with that legume (Bradyrhizobium leguminosarum) are also very slow growing (more than 30 hours as doubling time).
I want to add something for you; if you want to determine the best time to count the root nodules of a leguminous plant, you can do that at flowering stage of the respective leguminous plant.
In glass house experiments it is recommended minimum 3-4 weeks to visualize nodules after inoculation however in test tube experiments bumps on roots can be seen after 10 days of inoculation. Some wild plant took more than 2 months to visualize nodules in field conditions.
Root nodule formation by rhizobia depends on host species for example in soybean on which I work where the root nodule formation occurs at the flower initiation stage (about 32-35 days after germination).
"However, the bradyrhizobia associated with that legume (Bradyrhizobium leguminosarum) are also very slow growing (more than 30 hours as doubling time)"
In Common bean (Phaseolus vulgaris) you can see the nodules after 4-5 days when cultivated in growth pouches, however the complete size of well formed nodules and activity after 10 days. Commonly it iis differed depending on the legume host and Rhizobium species