Hey!
Congratulations on the offer!
In short: yes, it should be completely fine to email Bakers. They have explicitly stated that you can get in touch if you need an answer sooner, and this is a very normal reason to do so. It won't hurt your chances - if anything, it might show that you are a competitive applicant (as well as genuine interest in the firm).
I'd keep it polite and neutral, explain that you have received another offer with fixed dates, and ask if they have any updates on your application so that you can make a decision. Firms are generally aware that this is a possibility for candidates, as most schemes occur around the same time so there is bound to be overlap.
I would perhaps avoid pushing the first firm on dates again, as I'm sure they'll do all they can to accommodate your request