Do academics in the UK usually have to declare other jobs (e.g. consulting) they take on and is it public information? I am interested in relationships between academics at UK universities and private businesses. 

A number of academics in the Computer Science department at my old university worked part time and were also employed part time by google. I recently found out that a number of them also work a few weeks a year on a consultancy basis for a local finance firm.

I wondered whether 1) they had to declare any such activities to the department or university (especially in the consultancy case which seems to go alongside full time academic employment), and 2) whether it's possible for others to find out about these relationships (e.g. a journalist investigating links between an academic department and a particular company).
Regarding (1), whether they had to declare depends entirely on a) the agreement with private businesses (in particular, the agreement between the private business and the academic might actually be between the private business and the university) and b) the terms of the academic's contract. I believe that many contracts permit UK academics to work for private businesses during part of their time, but I don't know how widely that holds.

Regarding (2), it is always possible for others to find out about relationships. You could rephrase your question to: can a third party find out about these relationships legally? It then depends on the contract between the academic and the private business. In particular, if the contract contains a non-disclosure clause, then someone must violate that clause to enable a third party to discover the relationship, hence, it cannot be discovered legally.