I am literally in an identical situation.
Received a vac scheme offer for summer which I was able to negotiate prior to accepting my role which will start end of March.
I then received another vac scheme offer for April.
The new job starting in March is really good so I don't want to ask them for more time off already. I called the firm and explained that I want to accept the vac scheme for Spring, but only closer to signing my contract date will I be able to mention this to my new job employer. I then asked it, as a worst case scenario, I can't obtain the vac scheme, whether I can postpone it to next year. The recruitment team was exceptionally kind and understanding and said that they will allow it.
I think it's up to you what you feel comfortable with doing though!