The applicability of the MEL ends at V1 (or pick your definition of "takeoff"). After that, an equipment failure is handled according to the approved checklist. If there are no written restrictions, you can proceed as filed and shoot any required approach.
Most commonly there is a more restrictive equipment list for Cat II and Cat III ops. If you have a failure before DH or AH that brings the aircraft below the minimum requirements, you are obligated to go missed approach. As always, if there are other extenuating circumstances, the PIC can use his emergency authority to override those requirements.