There are many publications that are useful in informing expectations. In summary, they all suggest that you should see a downward trend in severity.
Onslow et al. (2002) found that there was a 30% decrease in average weekly severity ratings from clinic visit 1 to 5.

Jones et al (2000) did a large-scale file audit and reported that there was a median of 11 clinic visits to reach Stage 2 and 90% reached Stage 2 in 22 visits. Note that this was prior to the recommendation in the Lidcombe Program Guide that criteria should be met on three consecutive clinic visits before commencing Stage 2. Therefore 3 visits need to be added to the above number of sessions. Rousseau et al’s (2007) prospective study reported a median of 16 visits to Stage 2.

Koushik, et al. (2011) did a replication of Jones et al. (2000) for the North American population. Their file audit results replicated the Jones study with a median of 11 sessions to reach Stage 2, but the more severe the stutter the longer the treatment time.

Progress may be influenced by how successful treatment conversations with the child are and how regularly they are occurring. Sometimes, progress is first noticed in structured treatment conversations in that the child is able to say longer and more complex stutter free utterances. This then begins to generalise to reductions in daily severity ratings. It is worth noting that treatment times can be variable and children may respond after several sessions or it can take many months. In all cases if progress is not being observed then it is important to problem solve and consult with colleagues.

References
Jones, M., Onslow, M., Harrison, E., & Packman, A. (2000).Treating stuttering in young children: Predicting treatment time in the Lidcombe Program. Journal of Speech, Language, and Hearing Research, 43, 1440–1450.

Koushik, S., Hewat, S., Shenker, R., Jones, M., Onslow M. (2011) North-American Lidcombe Program file audit: Replication and meta-analysis. International Journal of Speech-Language Pathology, 2011; Early Online, 1–7

Onslow, M, Harrison, E, Jones, M & Packman, A (2002) ‘Beyond Clinic Speech Measures During the Lidcombe Program of Early Stuttering Intervention’ Acquiring knowledge in Speech, Language and Hearing, vol. 2, no. 2, pp. 82-85.

Rousseau, I, Packman, A, Onslow, M, Harrison, E & Jones, M (2007) ‘An investigation of language and phonological development and the responsiveness of preschool age children to the Lidcombe program’ Journal of Communication Disorders, vol. 40, pp. 382-397