Yesterday Boris Johnson gave a speech which has been claimed to be his resignation. At no point did he state that he had resigned either as the leader of the Conservative Party or as Prime Minister.
He did state that a leadership election for the Conservative Party would shortly be underway and that he would hand over to the new leader in the Autumn. Given that we now know that resigning ministers get three months pay it seems as if Boris is going to drag out the process so that he will be paid until the end of the year.
There are a number of investigations into his activities that need to take place (largely to do with his alleged improper Russian interactions). He cannot remain in office while they are happening. In any other setting he would be placed on suspension or put on gardening leave.
Boris Johnsons premiership has demonstrated holes in the British Constitution. Ministers are expected to act honourably and resign if their honour is called into question. This seems to have fallen out of practice. Given the weight that is given to president this cannot be left to stand.