Sometimes. There are three legitimate routes: agree a shorter notice period with your employer (the most common solution), accept payment in lieu of notice if they offer it, or walk out and breach the contract (rare, risky, but occasionally the right move). Each route has different consequences for pay, references, and the relationship. The easiest is almost always to ask.
The starting point is that your notice period is contractual. You committed to it when you signed the contract. The employer is entitled to the full period, just as you're entitled to the full pay and benefits during it. So 'leaving early' isn't a right; it's a negotiation. The good news is most employers will agree to a shorter notice when there's no operational reason not to.
The simplest route is asking. Tell your manager you'd like to leave earlier than the contractual date, explain why (new role start date, personal circumstances, agreed handover is complete), propose a specific new last working day. If they say yes, get it in writing (even just an email exchange) and the new date becomes binding. From the employer's side it's a request to vary the contract by agreement, which they can do freely.
Payment in lieu of notice (PILON) is the second route. If your contract has a PILON clause, your employer can pay you a lump sum equivalent to your salary for the unworked notice and end your employment immediately. This is more common when the employer wants you out (to a competitor, post-restructure, sensitive role) than when the employee wants to leave early, but employees sometimes negotiate it. PILON is fully taxable as earnings since the 2018 reforms.
Resigning and walking out (technically called 'wrongful resignation' or breach of contract) is the third route. You stop turning up before the notice period ends. The employer can sue you for damages caused by your breach. In practice this is very rare; the damages are limited to the cost of replacing you for the unworked period, and litigation against individual ex-employees is expensive and uncommercial. So nothing usually happens beyond a written warning on your file.
What can happen if you walk: your employer can refuse to pay you for any unworked notice (no salary for days not worked); they can refuse to give a positive reference; they may include a note about the breach on internal files visible to references; they may withhold contractual benefits like a pro-rated bonus. None of this is automatic; it depends on the employer's stance. Some are pragmatic and let it go; some make a point of it.
If you're moving to a regulated industry (financial services, regulated professions, public sector) the consequences of a contract breach can be more serious because future employers will see it through formal vetting. References may be required to mention contractual breaches, and new employer onboarding sometimes catches it. In those situations the agree-or-PILON routes are much safer than walking.
If something serious is happening (the role was misrepresented, harassment or discrimination, unsafe working conditions, unpaid wages) and you can't work the notice, the picture changes. You may be able to argue constructive dismissal or a fundamental breach by the employer that releases you from your obligations. Don't walk first and ask later; take advice from ACAS, Citizens Advice, or a solicitor before you stop turning up.
Practical recommendation: ask first. Most employers will compromise on a few days or a couple of weeks if you've handed over properly and the relationship is civil. Many will go further if your new employer is pushing for a fast start. PILON is the formal version if they agree but want a clean break. Walking is a last resort and should be a considered decision with advice, not a reactive one.
General information about UK employment law. For your specific situation, contact ACAS or an employment-law solicitor.