My experience of making such a change & the experience of friends and relations is that the old ISP will continue to accept incoming mails (and you can read them & reply by webmail at that ISP) for as long as your subscription there lasts. eg if you start your new BB account next Wed, 3rd July but you've already paid the previous ISP until the end of July then you've got nearly a month when you can still access emails at the old address. But if the previous was a pay-as-you-go it may last indefinitely. As already advised, tell everyone your new address asap and ask them to delete the old one from their address book.
As palinka says if it's an old pay as you go account then it may last indefinitely but you may have to use it to keep it alive ... if you want it kept alive. My old freeserve/wanadoo/orange account is still alive after nearly three years but it has to be used once every 90 days to stop it from being deactivated.
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