OK, this is kind of re-posting, but I think I can clarify the question more, so it's worth a shot.

Consider a real valued process $(X_t)_{t \leq T}$, cadlag on a probability space $(\Omega, (\mathcal{F}^\circ_t)_{t \leq T}, \mathbb{P}). \mathcal{F}^\circ_t=\sigma(X_s;s\leq t)$ is the uncompleted, natural filtration generated by $X_t$. Unfortunately $X_t$ neither has independent increments, nor is it markov. Since $\Omega$ is a Polish space, $\mathcal{F}^\circ_T$ and also $\mathcal{F}^\circ_t$ are countably generated, so we know, there exists a regular version of the conditional probability of $\mathbb{P}$ for any fixed $t$ for $\mathbb{P}$-a.a. $\omega$, i.e. for fixed $t$, $\mathbb{P}(\cdot|\mathcal{F}_t)(\omega)$ is a prob. measure f.a.a. $\omega$.

Hence we know, that for all $t\in [0,T]\cup \mathbb{Q}$, we find a regular conditional probability f.a.a. $\omega$, depending on $t$. In words: Given almost any path of the process up to time $t$, we can deduce the probablity of events, taking that information into account.

On the remaining $\omega$'s, define some meaningless measure, so we have a measure $\forall \omega$. How can I extend this to all $t$ in a reasonable way? Reasonable means: There is one Null set $N$, so that $\forall t$ $\mathbb{P}(\cdot|\mathcal{F}_t)(\omega)$, $\omega\in N^c$, is a measure Anybody seen anything like this?

I read something like this only for Markov and Feller processes using infinitesimal generators, but this cannot be carried over one to one, because we do not have a transition semigroup.

Maybe I have a deep misunderstanding here. Grateful for any objections, hints and comments.