Hey!
Congratulations on completing the vacation scheme, and for your existing TC offer!
Yes, I think that is perfectly appropriate and quite common in this situation - I had a similar situation last cycle when I was still waiting back post-VS for one firm, and had a TC offer from another.
I would email the graduate recruitment team and explain that you have recently completed the vacation scheme at the firm, which is your preferred choice, but you have another training contract offer that expires next week. You can then politely ask whether they are able to provide any update on the timing of their decision making process.
I would avoid framing it as a demand for an immediate decision, as they may still need time to complete their assessments. However, making them aware of your deadline gives them the opportunity to see whether they can expedite your application or provide some other further guidance.
I think it is important to bear in mind that they may not be able to accelerate the process, particularly if they are still making their way through the assessment process. However, I don't think there is any downside in asking politely and professionally, and at least you'll know either way. Graduate recruitment teams are used to candidates managing competing offers and deadlines, so I'm sure they've been asked similar questions before!
Wishing you the best of luck - I hope it works out for you!
