Can I refuse to take a test for drug or alcohol presence?

Your employer has no automatic legal right to test you for drugs or alcohol and you could refuse if this is not provided for your existing contract of employment.

Screening for drugs and drink is, however, likely to be a requirement in jobs where safety is of paramount importance, for example working in public transport and will, often, be a requirement indicated as an express term in the contract.

Where your employer keeps records of such tests, data protection rules apply and your employer must comply with these.