As above, it also sounds to me like you are talking about some manner of hierarchical modelling. I find that the book “Doing Bayesian Analysis” by John Kruschke has some nice examples of this, in particular chapter 9 quite insightful.
Just to be sure though, when you say you want to infer m1 independently on the first level, do you mean this level has its own likelihood that depends on m1 and level2 has some other priors that depend on m1 and its own likelihood and you want to sample these together? Or do you want to first use level 1 model independently to infer the posterior of m1 and then use posterior of m1 in the second model sampling it independently?
If the first case you can always model these two levels together as such:
m1 = prior(some fixed parameters)
m2 = prior(m1, some other fixed parameters)
likelihood1(D1;m1)
likelihood2(D2;m2)
Otherwise if you are talking about the latter case, you can try to include the derived posterior of m1 in the next model as normal distribution with posterior mean and sd (provided the posterior looks normal enough). To me the first way makes more sense though since likelihood of D2 depends on m2 which depends on m1, D2 could be used to further inform what are the likely values of m1 are. However there are I guess instances where the latter approach could make some sense too.