"The key to answering that question depends whether you overlooked a discussion over dates at the point where the offer was accepted? Make sure you always get some sort of agreement over dates right at the start, as otherwise, your conveyancing solicitor will not know how fast to work for you during their involvement with the conveyancing.
Often conveyancing solicitors are asked mid-way into the transaction by their own clients “how long will it be before Completion?”. Of course your conveyancer plays no part in the completion date, as that is the preferred date on which you – and only you – want to move.
What your conveyancing solicitor can say is how long they think it will be before they can exchange, but even then, whether they can exchange depends on when can any related buyer or seller exchange, and your conveyancer will not always know that. The estate agent will, as they 'chain progress' as part of their work – or at least the dynamic estate agents will – where they call lawyers along the chain. That is partly why they receive such high commissions, to spend time doing that and keeping the deal intact, while the conveyancing solicitors concentrate specifically on the legal work and make sure what you are buying or selling is legally sound.
However, the more dynamic conveyancing solicitors will also help over dates, of course they will. They are on your side, that is why you are paying them, as to be honest, they want to know when they will get paid, as a completion date means just that.
So what you and they might do in response to the question above is to at least circulate your preferred date for moving, and that starts the ball/discussion rolling.