The conversion will happen in 2023 and will be reported on your 2023 tax return. There is no retroactive option for conversions and rollovers, they happen when they happen.
The tax consequences of the IRA to Roth conversion depends on whether you have any pretax traditional IRA money in any account, even if it is a different account with a different broker. If you have pretax money, then your conversion is prorated and is not a true “back door“ Roth conversion. If you don’t have pretax money in an IRA, then you can do the back door conversion at any time. There is no time limit relative to the contribution.
