Between 1-2 seconds for the shifters to awaken.
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Canât help on the battery issue, but Shimano do allow the shifters to be updated wirelessly. Mine had a wireless update from their new in box state. However, I believe you need to be on the correct firmware to allow this in the first place. So if you arenât, then you need a wired update to enable future wireless updates. 
Which Shimano wireless shifters now allow firmware be wirelessly updated? I have done probably half a dozen firmware updates on the 12-speed Dura-Ace shifters - set up wireless - and all of them required the shifters be wired.
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After confirming with who is arguably one of the most knowledgeable resources on Di2 (bettershifting.com), I believe your âwirelessâ shifters are either set up wired, or they are not wireless shifters (the 12-speed rim brake Dura-Ace shifters are wired-only). The firmware on the current generation of 12-speed Shimano wireless shifters cannot be wirelessly updated.