Hi I am a new customer it states my date will be 13th but my tariff with sse ends today will my gas be cut off until the 13th?

No. You’ll be put on the variable tariff with SSE until your switch completes.

No, although sse may charge you a slightly higher rate until the switch date.
Just sit back as Bulb and sse will sort it out between them, nothing you need to do.