Assuming the Writ is moved today then Tuesday becomes Day 0 of the by-election process which can run for as short a period as 15 days and a maximum of 19 Days (Saturdays, Sundays, Christmas Eve, Christmas Day, Good Friday, bank holidays and any day appointed for public thanksgiving or mourning are not included in the Day Count).
If therefore today is day 0 then nominations would close not earlier than day 6 but not later than Day 8 and polling day would be fixed by the acting returning officer, between days 15 to 17, 16 to 18 or 17 to 19, depending on the day fixed as the last for the delivery of nomination papers. A by-election timetable is much more flexible than the general election timetable as the Acting Returning Officer has discretion in fixing the last day for the delivery of nomination papers and the subsequent polling day. Thursday has become the convention but there have been post war by-elections where polling day has been on another day.
To hold the poll on Thursday June 26th then that would be Day 17 and so Nominations would Open on Wednesday 4th June and could close as early as Day 6 Wednesday 11thy June or as late as Day 8 (Friday 13th June). Of Course that is not a minimum campaign. Polling Day could still be on June 26th if the writ was moved as late as Thursday this week. Day 1 would then be Friday 6th June and Nominations would close the next Friday (13th June).
The writ for a by-election is usually issued on the same day as or the day following a motion in the Commons for the Speaker to make out the warrant for the issue of a writ. There is a convention that the writ should be moved within about three months of the seat becoming vacant, but this is not a statutory or parliamentary requirement.
However, this is all academic at the moment since Boris Johnson has not actually yet resigned his seat. The procedure for that is for him to apply to the Chiltern Hundreds, an office under the Crown, the holder of which is barred from being an MP. The current holder of the Office is Tony Blair, since he used the same method to abandon his Sedgefield Constituency once it was no longer of use to him.
One thing no-one has been able to tell me is if the application to the Chiltern Hundreds must be made in person of if it can be done in writing? If so Boris can't resign until he gets back from holiday?