A confidence interval is a "post-data" interval estimate that is supposed to bracket the true parameter in %C of the samples. What you appear to be trying to do is to predict a future event. For this, you need a little more structure on your problem. In particular, a confidence interval is insufficient. What you really need is a distribution of possible head probabilities and probabilities of rolling 1. Then, you need to calculate the probability as such:
Let $C$ be the outcome of the coin toss and $X$ be the results of the die roll, $f_H(p)$ isthe density function on the probability of heads (i.e., $p$) and $f_{1|C=H}(q)$ is the density function for the probability of rolling a 1 (i.e, $q$) given that you got a head. Therefore,
$P(X=1)=E[1_{C=H}1_{X=1|C=H}]=E[1_{C=H}]E[1_{X=1|C=H}]$ where $1_{C=H}$ and $1_{X=1|C=H}$ are indicator functions that take value 1 when the event in subscript happens, and 0 otherwise. Conditional independence between the coin toss and die roll allow you to multiply expected values.
Now, $E[1_{C=H}]E[1_{X=1|C=H}]=\int\limits_0^1 \int\limits_0^1pf_H(p)qf_{1|C=H}(q)dqdp$. In other words, you multiply the expected values of the distributions on P(heads) and P(X=1|Heads). So, you will need more info to solve your problem as formulated.
IF you have the underlying data that produced each CI, then you can use methods from Bayesian prediction or predictive likelihood
