  In \cite{MbPe}, Mbombo and Pestov prove that the group of isometries of the
generalized Urysohn space of density $\kappa$, for uncountable $\kappa$ such
that $\kappa^{<\kappa}=\kappa$, is not a universal topological group of weight
$\kappa$. We investigate automorphism groups of other uncountable
ultrahomogeneous structures and prove that they are rarely universal
topological groups for the corresponding classes. Our list of uncountable
ultrahomogeneous structures includes random uncountable graph, tournament,
linear order, partial order, group. That is in contrast with similar results
obtained for automorphism groups of countable (separable) ultrahomogeneous
structures.
  We also provide a more direct and shorter proof of the Mbombo-Pestov's
result.
