The J1 grace period is a matter arising under US law only and not recognised outside of US jurisdiction. The UK Border Force will use the starting and ending datesdate on your J1 to determine if you qualify for visa free transit. Based upon what you wrote, you will not qualify because you will not have a valid exemption document.
It means you will need a UK transit visa, or alternatively to change your schedule so that your J1 can be used as an exemption document.
Overall, you cannot rely upon 'grace periods' and other concessions that arise outside of UK jurisdiction. NOTE: the transit visa rules may be changing between now and your travel dates.