The revised quotation should follow the same process and timescales that would apply if the compensation event was being dealt with at an earlier point in the programme. The fact that this means that the compensation event will be implemented after either Completion or the Completion Date is irrelevant.
Only consider dispute resolution as a last resort, not a first! Consider using it if you don’t get what you want by using the compensation event procedures.
If you do decide to use dispute resolution W1 doesn’t permit referral to an adjudicator at any time, W2 states this in order to comply with HGCRA 1996. Under W1 you must comply with the time limits imposed by the Adjudication Table. Note that the table makes no reference to the terms Completion or Completion Date, it refers to timescales applying from notification of the dispute to the other Party.
So in simple terms, yes, you can refer dispute after either Completion or the Completion Date.