An EIC is required for the new circuit and chargers consumer unit, Part P notification and DNO notification as per Code of Practice.
The EIC can be hard copy or electronic copy, which is not always supplied on the day.
Part P notification to your councils building control dept, can be checked on your councils online planning portal if you don’t receive a copy, however one should be sent out within 30 days. (I assume this equally applies to NI)
The installer will notify the DNO of the connection, you can check its been notified correctly correctly, if you like by ringing them in some cases your supplier may send a letter out, although I wouldn’t worry about this.
The certification goes to the owner of the building, so if your in rented housing, it will be forwarded to your landlord 🙂