Looking at the title of BIP 341 (Taproot: SegWit version 1 spending rules), does that mean we'll see address with prefix bc1p?
yes. in a Bech32 encoding when you set the witness version to 1 the first character after the separator (ie. 1) is going to become letter "p".
BIP 173 doesn't mention this but it is easy to use one of the libraries to encode an arbitrary length byte array to see what the first character is.
https://github.com/bitcoin/bips/blob/master/bip-0173.mediawikiempty 32 bytes= bc1pqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq5us4ke