This is a pretty common/standard terms in any employment contract for full time roles, especially in regulated industries or the public sector.
Enough people breach their employment contract by undertaking various forms of employment on top of their job that stipulates this type of term.
How seriously your original employer will consider it a breach of contract is going to depend on how much it conflicts with or impacts your job. There can also be some issues of honesty and ethics if you have not been truthful about the situation.
However, many vacation scheme students manage to keep it quiet and no one finds out. When they do, they often have to think about what they say to various people - not just in their current job but also the people they will interact with during their vacation scheme.
Being tagged on LinkedIn or having your photo on a “welcome our vacation scheme students” post is a possibility - so you have to factor that risk in. How connected your job/employer is to your vacation scheme firm means that risk is going to be varied from person to person.
You’ll also have to factor in conflict checks too. It’s not impossible for a conflicts check to flag up something that would potentially need to be flagged back to your employer or that the firm would say you would need to flag to your employer. But again the chances of this highly depend on what your job is and who you are employed by.
But ultimately you are going to breach of your employment contract and you have to consider that doing so could lead to your job ending. You have to calculate your risks and your risk tolerance to work out what to do.
