The turbine engines for the Britannic were indeed manufactured by Harland and Wolff. However, when the Olympic class was first contemplated (and I suppose we should also include the Laurentic in this as well) Harland and Wolff did not have the required experience or technology to design and manufacture marine turbines. The main reason for this is that turbine technology was still relatively new, and with few military contracts (the largest potential user of marine turbines), there had been little incentive to develop such expensive facilities at Belfast.
As a result the company's early turbine work was contracted out to John Brown & Co. of Glasgow. Each company purchased a number of shares in the other (basically a merger) and upon obtaining the appropriate licenses from the Parsons company the turbine work was sub contracted to Brown's (on a commission basis) until the required facilities and expertise had been established at Belfast.
The reciprocating engines, of course, continued to be built in Belfast.
Hope this helps.