First, to clarify some of the issues raised in the question:
The election in a constituency ends when the Returning Officer declares the result. The person declared the winner becomes the Member of Parliament for that constituency. (While election results can be challenged after that, the grounds for doing so are quite narrow, so successful challenges are rare.)
Specifically, an MP's salary begins "the day after the day of the poll for the parliamentary election at which the member was elected" (Parliamentary Standards Act 2009, s4(5)).
But: "No payment of salary is to be made to a member before the member has made and subscribed the oath" (s4(6)).
This seems absurd to me since one clearly would not and should not be able to vote on bills, respond to casework or otherwise assume the duties and privileges of office until the vote tally is known.
To take the 2024 election as an example, the election took place on Thursday 4 July. Most of the results were known by the following day, with a couple of close results announced the day after that.
Parliament met for the first time the following Tuesday (9 July), for the Commons to elect their Speaker. (No vote took place this time, as Sir Lindsay Hoyle was re-elected without objection; at other times, there could have been an election for a new Speaker.) With that done, MPs then began the swearing in process, which took about a week.
The State Opening and King's Speech took place on Wed 17 July, with the debate on that lasting a few days.
The first opportunity for MPs to vote on anything took place on 22 July, that being a vote on an amendment to the King's Speech. The first vote on a bill was on 29 July - 25 days after the election, and 23 days after the final results were declared.
So the idea of MPs not being able to vote "until the vote tally is known" is moot. Even if there is a Speaker election, the general election results are determined a few days before MPs first meet in Parliament.
As for responding to casework, etc: while it takes time for MPs to move to London (for those who don't live locally), set up an office, and deal with other admin, casework continues to come in, and it appears that new MPs do get down to business as soon as they're able.
whether the taking of the oath of office and the assumption of office might perhaps be considered the same
They are not. As mentioned above, a candidate becomes an MP when their result is declared (and no earlier than the day after the election). However, they are not eligible to participate in Commons proceedings (other than the election of the Speaker), nor - crucially - to receive a salary, until they swear. Notably, Sinn Fein MPs refuse to swear allegiance to the British Monarch, so they never appear in the Commons chamber - but they are nonetheless MPs, and still deal with casework and other MP business outside the chamber.