Re: @Gordon I am not
"Does this mean that the "public" get a different IP address?"
Short answer: Yes
IP you get on BT WiFi's public unencrypted connection is completely unrelated to the IP range you have on your private, encrypted connection. Each gets NAT-ed and passed to the exchange separately from different IPs.
Additionally, BT WiFi is authenticated after connection, so even if 10 people are connected to the same public hotspot, each MAC/IP address is non-anonymous. And public access via your BT router also doesn't use up your data allowance if you are on a metered deal, due to the same non-anonymous, non-plausibly-deniable nature of the service where everything is completely logically separate even if it is multiplexed over the same physical wire.
So - no anonymity, no plausible deniability.
The physical bandwidth shortage _can_ be an issue since the connection is generally limited by the sync speed, but in reality it is very unusual to see prolonged heavy impact from this. And if you do see an impact from it, you can always switch it off, and in the process forego your own access to BT WiFi hotspots.