  We show that for groups acting acylindrically on simplicial trees the $K$-
and $L$-theoretic Farrell-Jones Conjecture relative to the family of subgroups
consisting of virtually cyclic subgroups and all subconjugates of vertex
stabilisers holds. As an application, for amalgamated free products acting
acylindrically on their Bass-Serre trees we obtain an identification of the
associated Waldhausen Nil-groups with a direct sum of Nil-groups associated to
certain virtually cyclic groups. This identification generalizes a result by
Lafont and Ortiz. For a regular ring and a strictly acylindrical action these
Nil-groups vanish. In particular, all our results apply to amalgamated free
products over malnormal subgroups.
