The ESB's - and all other supplier's - billing periods are every two months.
If your meter has just been read and Bord Gais have yet to switch you over to their system you will get your normal bill from the ESB any day now.
Whether Bord Gais will eventually switch you over from the reading and date you supplied (they should) or if they will use the reading just obtained by ESB Networks or if they will wait another two months for the next reading is anyone's guess.
The situation is becoming increasingly confused. I would phone Bord Gais and establish exactly what they intend to do as regards your switch. Being told that it could take up to 60 days is not helpful. What you need to know is what date and reading you will be switched from as this will determine if you need to pay your next ESB bill or not.
Regards,
Fnergg