You can pause a streak after any win and come back at the same point (possibly even changing teams for next time, as it will still count as the same streak), but you do not get those items until you lose to break the streak (or, in non-Super formats, when your streak is forcefully terminated after battle 20). Even after that, though, you have to wait a day, until midnight rolls around. Then the next time you try to enter the Battle Tree challenge, the attendant will give you the consolation items first.
Getting a consolation item doesn't mean you lose out on the previous items on the list, so losing a streak of at least 5 means you will get Moomoo Milk among the consolation items, no matter how long after 5 you kept going to get more items further down the list. The last items on the list are Lansat Berry (for extending your streak to at least 100 before losing) and Starf Berry (for extending to at least 200).