The issue here is that if this person has been continuously employed at their place of work for less than 2 years then their employment rights will unfortunately be somewhat limited. Most importantly, they will not be protected against unfair dismissal. This means that their employer can dismiss them for more or less any reason, and without following a fair procedure, as long as their decision is not based on discriminatory grounds (i.e. because of gender, race, religion, age, a disability, sexual orientation, etc.) or because he was trying to assert any of his statutory rights (e.g. requesting paternity leave, etc.).
If the dismissal had nothing to do with any of the above exceptions then they would not be able to challenge it and their only protection would be if they were not paid their contractual notice period, because unless they were dismissed for gross misconduct, they would be entitled to receive their contractual notice period. This is not gross misconduct because the conviction happened after they took the job so unless they specifically failed to mention this when they had a contractual obligation to do so, it would not amount to gross misconduct.
Therefore, a dismissal is indeed possible I the employer wants to go down that route. In terms of whether they would be notified, they would not be notified by the DBS unless a new DBS check is made.
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