No, not while you're still employed. Your existing contract continues until the last day of your notice period (whether you're working it, on garden leave, or off sick). Starting a new role before then breaches your current contract and almost certainly breaches the new one too. The simple rule: one employer at a time. If you want to start sooner, negotiate a release first.
Every UK employment contract carries implied and usually explicit duties of loyalty, fidelity, and confidentiality. Working for a second employer in parallel breaches all three, particularly if the new role is in the same industry. Even if the new role is in a completely different field, most contracts contain an 'exclusivity of service' clause that requires you to devote your full working time to your current employer until your employment ends.
Garden leave is the trickiest because to outsiders it can look like you've already left. You're not in the office, you're not visible at meetings, your email may be auto-replied. But the contract continues; you're still receiving full salary; you're still bound by every obligation. Starting a new role during garden leave is one of the more dangerous breaches because the new employer is also taking the risk of inducing your breach. Most won't.
What you can do during the notice period: anything that isn't paid work for another employer. Personal projects (not commercial). Learning, training, certification. Reading. Talking to your new manager about logistics. Helping family and friends in unpaid ways. Charity or volunteer work that doesn't compete. Travel, exercise, family time. None of this conflicts with the contract.
Some employers will release you early if you ask. This usually happens via PILON: you stop being employed immediately, you receive the unworked portion of your notice as a lump sum (taxable as earnings), and you're free to start the new role the next day. Whether the employer offers this depends on whether their contract has a PILON clause and whether they want you off payroll quickly. Junior roles rarely get it; senior or competitive exits often do.
The new employer's perspective matters. Most reputable employers won't ask you to start before your previous notice ends. The legal risk to them (inducing breach of contract) plus the optics (looking like they don't respect contracts) plus the practical risk of you carrying contamination from the previous role mean they'd rather wait. If they push you to start before your notice ends, that's a flag worth pausing on.
Regulated industries (financial services, accountancy, law, regulated professions) have additional formal vetting that catches contract breaches between roles. New employer onboarding in those industries includes confirmation from your previous employer that you've left cleanly. Starting before the notice ends will be caught. The reputational and regulatory cost outweighs almost any short-term benefit of starting a week early.
If the new role has a hard start date that doesn't fit your notice period, negotiate the start date rather than the notice. New employers are usually flexible about delaying a start by a few weeks once they understand the constraint. They've seen this before. Pushing back a start date is administratively trivial; navigating a contract breach is messy. The cleaner path is to wait.
Practical recommendation: confirm your last working day with your current employer in writing, agree the official termination date, then negotiate your new role's start date for at least the day after. Build in a few days of buffer if you can; transitions are stressful and a clean break is worth more than the lost income.
General information about UK employment law. For your specific situation, contact ACAS or an employment-law solicitor.