The trial basis issue would not really be relevant. If there is a written contract in place and it contains a specific clause detailing the notice period an employee is supposed to give if they wanted to leave their employment, they will be contractually bound by it. Therefore, if the employee fails to honour this notice period then they will be acting in breach of contract. The employer then has the option of suing the employee to seek compensation for damages resulting from their breach. However, in reality such claims are very rarely made. This is mainly due to the costs and time involved, also the relatively small damages that can be recovered. Also the employer has to show that actual losses have been incurred and often that is not easy to do. So whilst there is no way of predicting whether the employer will take this any further or not, chances are that they will not. A more likely outcome is that the employer refuses to provide a reference in the future or if they do, it could mention that the employee had breached their contract.
It is therefore best to try and negotiate a mutually acceptable notice period that would suit both parties. However, if that is not possible and there is a pressing need to leave early, that is still a possibility, subject to the risks identified above.
Finally, there are circumstances when an employee may be entitled to leave with immediate effect and without honouring their notice period. This occurs when an employer has committed a serious breach of contract first. The whole contract, including the notice periods, then becomes immediately void and the employee would be treated as being 'constructively dismissed'. So if there are reasons to believe the employer has acted in breach of contract, whether a breach of an express contractual term, or other breaches such as bullying, exposing the employee to unreasonable stress, discriminating against them, etc this reason can be relied on in order to leave with immediate effect.
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