Let's say Bruce is CFO at Exxon Mobil and his brother Clark works under the CTO at BP.
Considering possible implications of the fact that these brothers are expected as siblings to have a reasonably close relationship, would this scenario be considered illegal?
... however, both must be careful to uphold their duties (as directors) to the shareholders of their respective companies.
Bruce and Clark need to be particularly careful about many of these duties. The most relevant, given the generality of your question, is the duty to avoid conflicts of interest (s175 Companies Act 2006), which states that directors must avoid situations in which they have, or can have, a direct or indirect interest (relating to arrangements that don't directly involve the company – for those that do involve the company see, eg, s177).
Applying the above law to the facts of your question, even if Exxon decides not to enter into a contract with a given client, Bruce would be in breach of this duty if he suggests that Clark / BP can do the work instead.