You can't have BE and Infinity at the same time.
But sometimes you order Infinity and get an install date; but the installer doesn't turn up. (See lots on the BT forums)
They then reschedule, often for a couple of weeks later or more.
If you have arranged for your BE contract to end exactly on the Infinity install date, you can end up with no broadband for a while.
You might be able to arrange BT ADSL pretty sharpish in that case; but with complications because the BE LLU might mean work at the exchange to arrange that.
Also, BE are apparently helpful about short extensions to end of contract, but there can be details (line cease or ???) which can mean they can't help even if they want to.
Luckily, this sort of problem is fairly rare, but as I said, not as rare as it should be ....
I think my Infinity install was around 3 weeks after order: I can't remember now.
They do sometimes install at weekends; I don't know how difficult it is to get a weekend appointment, and whether it might mean a longer wait.
Moved (with trepidation turned relief) to BT Infinity 2 for upload speed. Happy BE user for several years.