Thanks for your question.
I regret to say that it is not possible for one person to attend the Registry office on behalf of another in order to obtain a Certificate of No Impediment to Marriage. One would usually make an appointment and take photographic proof of identity (eg. Passport) and address (utility bill or council tax demand/invoice) and then the CNI would be provided
You should though contact the British Embassy or consulate nearest to where you are to marry and ask them to provide you with their details on how to apply for a CNI from abroad. You usually can and they should be able to provide you with the appropriate application forms. You should also check with the Ukrainian authorities whether they accept a CNI obtained in this way, whether they require it to be legalised by the UK Legalisation Office and/or whether they require it to be translated in to the local language by the Institute of Linguists.
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