A representation of the Birman-Wenzl-Murakami algebra BW_{ t}(-q^{ 2n},q) exists in the centraliser algebra End_{ U_q(osp(1|2n))}(V^{ \otimes t}), where V is the fundamental (2n+1)-dimensional irreducible U_{ q}(osp(1|2n))-module. This representation is defined using permuted R-matrices acting on V^{ \otimes t}. A complete set of projections onto and intertwiners between irreducible U_{ q}(osp(1|2n))-summands of V^{ \otimes t} exists via this representation, proving that End_{ U_q(osp(1|2n))}V^{ \otimes t}) is generated by the set of permuting R-matrices acting on V^{ \otimes t}. We also show that a representation of the the Iwahori-Hecke algebra H_{ t}(-q) of type A_{ t-1} exists in the centraliser algebra End_{ U_q(osp(1|2))}[(V^{ +}_{ 1/2})^{ \otimes t}], where V^{ +}_{ 1/2} is a two-dimensional irreducible representation of U_{ q}(osp(1|2)).
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