Personally, I don't think there is any way that Heathrow can continue with single runway operations until October. Traffic is starting to pick up, as has been commented on by others, and current slot projections suggest the airport will be handling more than 600 daily flights before the end of August. Of course the situation is still very fluid, and there is no guarantee that operators' current plans will come to fruition, but it would be a big gamble to commit to an extended period of single runway operations just as the recovery is gathering pace. No airport, even Heathrow, wants to find itself in a position where it is turning away traffic because it is unable to meet demand.
With British Airways consolidating all of their European operations at Heathrow until the end of the summer and several other carriers that normally operate to Gatwick securing Heathrow slots, I suspect assurances may have already been given to airlines that both runways will be available for the remainder of the summer season once traffic has reached a certain level.
I can understand why such an idea was under consideration when traffic collapsed in late March and early April, but I think that boat has already sailed.