Clarification
The way I understood your question was:
- Participants will collaborate in sets $(P_1, P_2, \ldots)$ of $t+1$ participants each, and reconstruct the secret.
- They will keep doing this, until every participant has learned the secret (at least once)
- The question then is to find bounds for the number of required distinct sets $P_i$. In words: "How many different groups of participants are required (at most/at least) such that every participant learns the secret"
Lower bound
There will be a total of at least $\lceil\frac{n}{t+1}\rceil$ sets of $t+1$ participants each, reconstructing the secret. At least two of these sets will have a non-empty intersection, unless $t+1$ divides $n$, in which case a pairwise disjoint split would be possible.
Upper bound
On the other hand, an upper bound for the number of distinct sets of $t+1$ participants each, such that every participant would learn the secret at least once, would be given by $n - (t + 1) + 1$.
Aside
Of course the premise is of questionable use. Naive reconstruction only works in a setting with no active adversaries, in which case you might just as well have the first group which reconstructed it broadcast the secret.