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I've been issued T2 General visa on 01-May-2019 with a validity from 27-May-2019 to 23-Jun-2022.
I entered UK on 05-Jun-2019 to start employment a week later.
Renewal was applied in May-2022 and Skilled Worker visa was issued Valid from 09-Sep-2022 for 3 years.

When can I initiate the ILR process? Should I start only from 27-May-2024 or 28 days before i.e. 30-Apr-2024?

Also, do I have to take the Life in the UK Test? I've already got my UK NARIC validation during the T2G process.
My Wife may still have to take the exam as a dependent. Is that correct?

It is counted from the first visa issue date. Anyone who applies for ILR must pass the Life in the UK test. You can use your old NARIC certificate for the English language, if it was obtained under the Red route. Your wife needs to prove her English at min B1 level using a SELT exam