Algorithm for a special case of SAT/#SAT
From Solving #SAT using vertex covers, published at SAT'06 by Naomi Nishimura, Prabhakar Ragde, and Stefan Szeider:


  A cluster formula is a variable-disjoint union of so-called hitting formulas; any two
  clauses of a hitting formula clash in at least one literal. The known polynomial time
  algorithm for computing the number of models of a hitting formula can be extended in a straight-forward way to compute the number of models of a cluster formula.


Clash is later clarified/defined as:


  We consider propositional formulas in conjunctive normal form (CNF), represented
  as sets of clauses. That is, a literal is a (propositional) variable $x$
  or a negated variable $\overline{x}$; a clause is a finite set of literals not containing a complementary pair $x$ and $\overline{x}$; a formula is a finite set of clauses. For a literal $ℓ = \overline{x}$ we write $\overline{ℓ} = x$; for a clause $C$ we set $\overline{C} = \{ \overline{ℓ} : ℓ ∈ C \}$. We say that two clauses $C$, $D$ overlap if $C ∩ D \neq ∅$; we say that $C$ and $D$ clash if $C$ and $\overline{D}$ overlap. Note that two clauses can clash and overlap at the same time. [...] A formula is a hitting formula if any two of its clauses clash (see [17]). A cluster
  formula is the variable-disjoint union of hitting formulas [...]
  
  Lemma 3. #SAT can be solved in polynomial time for cluster formulas.


So their cluster formula seems to be exactly what you have defined.

There's also a journal version that paper, it seems; the result is (not surprisingly) also mentioned in a 2011/2012 survey paper "Backdoors to satisfaction" on which Szeider is a co-author, and which was published in some festschrift My first instinct that this was a more suitable question on cstheory.SE was perhaps not wrong. :-)

Also the notion of hitting formulas is cited to [17]:


H. Kleine Buning and X. Zhao. Satisfiable formulas closed under replacement.
In H. Kautz and B. Selman, editors, Proceedings for the Workshop on Theory
and Applications of Satisfiability, volume 9 of Electronic Notes in Discrete
Mathematics. Elsevier Science Publishers, North-Holland, 2001.


In another paper, the fact that


  It is known that for hitting formulas the satisfiability problem can be solved efficiently [7].


is cited to an even older paper:


K. Iwama "CNF satisfiability test by counting and polynomial average time" SIAM J. Comput., 18 (1989), pp. 385–391