No, generally not. Garden leave keeps you legally employed by your current employer, so all the standard contractual obligations still apply: no working for competitors, no consulting in the same field, no contacting clients, often no work for any second employer at all if your contract has an exclusivity clause. Completely unrelated paid work (driving, hospitality, family-business help) is usually fine but check first.
The reason garden leave exists is to keep you out of the market during your notice period while you remain a contracted employee. That value evaporates if you can take a second job in the meantime. So the standard position in UK employment contracts is that you can't work for anyone else during any part of your employment, garden leave included. The clause is usually called 'exclusivity of service' or similar.
What counts as 'work' for this purpose is broader than people expect. A salaried role at a competitor is obviously out. So is freelance consulting in the same field, even if it's unpaid. So is starting a business that would compete with your current employer. Less obviously, regular paid work in any field can be caught by a strict exclusivity clause; the test is whether your contract allows external work at all, not just competitive external work.
Working for a competitor specifically is the highest-risk scenario. Even if your contract has no garden leave clause, every employment contract has implied duties of loyalty and confidentiality. Joining a competitor while still employed breaches both. The new employer is usually unwilling to take that risk too; if you're moving to a competitor, they will almost certainly want you to wait until your employment formally ends before starting.
Completely unrelated paid work tends to be tolerated in practice. Driving for a delivery app, helping out at a friend's bar, working a few hours at a family business, doing volunteer or charity work: these don't compete with your employer and don't risk confidential information. But the contract is still the contract. If yours has a blanket exclusivity clause and the employer wants to enforce it, they can. Best practice is to ask in writing and get permission in writing.
Studying, training, and self-development are not 'work' in this sense and are usually entirely fine during garden leave. So is reading widely in your field, taking courses, writing for yourself, and personal projects that aren't commercial. Garden leave is often actively used by people for skills they wouldn't otherwise have time for; that's part of why it's seen as the most generous of the exit modes despite the restrictions.
Preparing for your next role is fine in principle but tricky in practice. You can read publicly available material about your new employer, learn about their market, talk to people informally, and plan logistics. You can't do any actual work for them, can't be paid by them, can't be on their payroll, and can't represent them externally. A short coffee with your new manager to confirm start dates and logistics is fine; turning up to a strategy day isn't.
If you're tempted to take a short freelance gig during garden leave because the financial difference is significant, weigh it against the risk. Garden leave breach is a clear breach of contract; the employer can terminate without notice (losing you the rest of the salary and benefits) and could in theory sue for damages. In practice the worst-case scenario is rare, but the contractual position is unambiguous. Get permission, or wait.
General information about UK employment law. For your specific situation, contact ACAS or an employment-law solicitor.