Depends on the signage, local laws and the reason why the road was originally marked as oneway:bicycle=no. Any oneway with a separate contraflow cycleway in an area where cycling against oneways is generally forbidden very likely won’t have an explicitly signposted exception for bicycles anyway, so the only reason why it would be tagged as oneway:bicycle=no is if the cycleway is not mapped separately and is combined with the road.
In a case like this, the oneway:bicycle=no tag could be removed from the main road when the separate cycleway is added because it only existed in the first place because of the (incomplete) mapping.
However, this is not valid if
- local laws generally allow cycling against oneways unless explicitly forbidden (unsure if this exists)
OR - cyclists aren’t required to use the separate cycleway and could also use the main road (explicit or implied
bicycle=optional_sidepath)
Since this question originally came up because of StreetComplete, I am strongly against removing/discarding oneway:bicycle=no in a (semi) automated way.