It is difficult to see what is going wrong without the data counts.

However in the Y ~ BB(n,mu,sigma) distribution model, Y|p ~ BI(n,p) where I believe p ~ BE(mu, [sigma/(1+sigma)]^0.5). [Note that sigma lies between 0 and infinity, while [sigma/(1+sigma)]^0.5 lies between 0 and 1.]

So in your intercept model, p ~ BE(0.501,0.328) which may be more reasonable.

To sort out the error messages from your model with time, I suggest you email the developers of gamlss directly, ideally with the data counts.

It is difficult to see exactly what is going on without the data counts.

However in the Y ~ BB(n,mu,sigma) distribution model, Y|p ~ BI(n,p) where I believe p ~ BE(mu, [sigma/(1+sigma)]^0.5).

[Note that sigma lies between 0 and infinity, while [sigma/(1+sigma)]^0.5 lies between 0 and 1.]

So in your intercept model, p ~ BE(0.501,0.328) which may be more reasonable.

For your model with time as explanatory variable:

Note that as sigma → 0, then BB(mu, sigma,nu) → BI(n,mu).

So for those time points with a very small fitted sigma, this suggests that a BI(n,mu) distribution model for the response variable is adequate [rather than a BB(mu, sigma,nu) distribution model, which is an over-dispersed BI(n,mu) distribution].

This seems a very reasonable conclusion to me (although I agree that the very small sigmas look odd at first sight).

I think the warning messages are just warning you that some of the fitted sigmas are very low.

I suggest that you plot the fitted mu against time, the fitted sigma against time, and the fitted [sigma/(1+sigma)]^0.5 against time.

If you still have problems, post them, or email the developers of gamlss directly ideally with the data counts.