Your manager, and consequently the company, are in breach of the Equality Act 2010 and they need to be made aware of that and the possible consequences if they don’t step up and tackle your utterly ignorant and self centred colleague - who, by the way, is not being discriminated against either as a vegetarian or a mother. This would only be the case if she was the only one being asked not to eat nuts in the office.
OP your allergy qualifies as a disability under the Equality Act 2010 and as such your employer has to make reasonable adjustment so that your condition does not put you at a disadvantage in the workplace. It seems as though they have already done this for you and another colleague and the reasonable adjustment here is simply making your office nut free and putting up warning notices to that effect. This is protected in law and therefore your colleague can’t simply rock up and eat nuts in a protected workspace.
Apart from the obvious consequences to your health, which I think breach health and safety at work laws, there are consequences under the Equality Act if your employer does not resolve the situation - your manager has a responsibility to do this, he can’t simply leave it for you to ‘sort out among yourselves’.
You have what’s called ‘protected characteristics’ under the Equality Act, by reason of your disability, and that means that you do not need to have worked for your employer for the statutory two years to be able to take action if they refuse to resolve a problem directly related to your disability. You should also know that a probation period has no meaning in employment law - it doesn’t absolve the employer of the responsibility to treat you fairly, or of the obligation to honour notice periods.
Your employer cannot dismiss you for anything disability related unless they can show that they have tried their best to make reasonable adjustment to accommodate it. In this case it’s blindingly simple - they keep the office nut free and the problem is solved. By allowing an employee to breach the protected workspace created for you and your colleague, they have broken the agreement and must put it right. Dismissing you because they don’t want to upset another employee, isn’t an option and if they were to give you notice you would potentially have a case for unfair dismissal.
If they refuse to resolve the situation and the working environment continues to cause you problems to the point where you have to leave, this is called constructive dismissal - they have made it impossible for you to continue working there. Again you would potentially have a case for unfair dismissal on those grounds.
The simplest way to resolve this is to ask for a meeting with your manager, and ask for someone to accompany you - a union rep or colleague - to take notes. Then make the manager aware of their responsibilities under the Act and ask formally for the situation to be resolved. You can find a copy of the Equality Act 2010 online here https://www.legislation.gov.uk/ukpga/2010/15/contents. I would suggest specifically familiarising yourself with the sections on reasonable adjustment and protected characteristics. ACAS can also advise you accordingly but they won’t approach an employer on your behalf. If your manager is willing, they can approach ACAS who will advise directly in those circumstances.
You do not need a formal diagnosis of a condition to qualify as disabled under the Act - the disability or health condition just has to meet the eligibility criteria for you to receive the protection described above. Obviously if you have medical evidence to support what you’re saying then it’s a good idea to take it along and have it noted in the meeting. Good luck.