Can a contract override the Construction Act?

No. The Construction Act, the Housing Grants, Construction and Regeneration Act 1996 as amended, sets a floor of rights for most construction contracts, and where a contract falls short the Scheme for Construction Contracts, a standard set of fallback rules, is read in automatically to fill the gap. That is why a contract cannot take away your right to adjudication, the fast 28 day dispute process the Act guarantees, your right to interim payments, the staged payments due as the work goes along rather than only at the end, the notice rules around those payments, or your right to stop work if you are not paid, which comes with its own written notice steps. What drafting can still do is obey the letter of the Act while working against you in the detail, through tight timings, extra conditions and payment mechanisms that are lawful but loaded in the other side's favour. So compliant and safe are not the same thing: check how the machinery runs, not just that the required words appear.

Say a joinery subcontractor signs a contract with a clause stating there is no right to refer disputes to adjudication, everything must wait for the courts. Months later they are owed £30,000 and want the quick 28 day process. The main contractor points to the clause and says they gave that right away. That clause is likely to be ineffective, because the Act guarantees adjudication and a contract cannot strip it out, so the Scheme would usually step in and the subcontractor can still refer the dispute. It was trying to remove a right the law protects, exactly the kind of term the Scheme quietly overrides.