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I don't have access to my company laptop and thus can't access this 'portal'.

Can I resign by emailing my resignation to the general HR contact, and cc my direct line manager? Will I be in breach of contract because the firm may say I actually never resigned because I didn't follow the procedure? It is physically impossible for me to resign in 'right' way since I don't have my company laptop.

I am not a lawyer. As others have said, talk to Citizens Advice or ACAS - but you cannot be prevented from resigning as that would be slavery.

With that it of the way, my advice: don't do it via email. Do it via an actual physical letter, signed using an actual pen with your actual signature stating that you "resign with immediate effect". Send this recorded delivery to the registered address of your employer as listed at Companies House - this is important as letters accepted at the company's registered address are legally deemed to have been read. Send a copy by email as well, but it's important to establish the paper trail - that means there can no dispute as to whether and when you resigned.

Then of course work your notice period in the most professional manner possible.

The UK government's website only talks about whether a company can require written notice instead of merely verbal notice. This seems to be written on the assumption that written notice is always sufficient.

Giving notice

You must give at least a week’s notice if you’ve been in your job for more than a month.

Your contract will tell you whether you need to give notice in writing - otherwise you can do it verbally.

Give written notice if you think you’ll need to refer to it later, for example at an employment tribunal.

There is a free Acas helpline where more qualified people can give you legal advice on your rights and responsibilities in this regard. But UK employment law is still fairly protective of workers rights, so I'd be astonished if a company could legally make it impossible for you to resign.