We consider a three-dimensional lattice $U(1) \times U(1)$ 
and $[U(1)]^N$ superconductors in the London  limit, with
individually conserved condensates. The  $U(1) \times U(1)$
problem, generically, has two types of intercomponent
interactions of different characters. First, the condensates
are interacting via a minimal coupling to the same
fluctuating gauge field. A second type of coupling is the
direct dissipationless drag represented by a local
intercomponent current-current coupling term in the free
energy functional.  In this work, we present a study of the
 phase diagram of a $U(1) \times U(1)$ superconductor which
includes both of these interactions.  We study phase
transitions and two types of competing paired phases which
occur in this general model: (i) a metallic superfluid phase
(where there is order only
in the gauge invariant phase difference of the order
parameters), (ii) a composite superconducting phase where
there is order in the phase sum of the order parameters
which has many properties of  a single-component
superconductor but
with a doubled value of electric charge. We investigate the
phase diagram with particular focus on what we call
``preemptive phase transitions.'' These are phase
transitions {\it unique to multicomponent condensates with
competing topological objects}.  A sudden proliferation of
one kind of topological defects may come about due to a
fluctuating background  of topological defects in other
sectors of the theory. For $U(1) \times U(1)$ theory with
unequal bare stiffnesses where components are coupled by a
non-compact gauge field only, we study how this scenario
leads to a merger of two $U(1)$ transitions into a single 
$U(1) \times U(1)$ discontinuous phase transition. We also
report a general form of
vortex-vortex bare interaction potential and possible phase
transitions in an $N$-component London superconductor with
individually conserved condensates.