Hi Andy --- Declaring bankruptcy in itself will have a long term detrimental effect on you financially, the missing payment on your credit card will be negligible compared to the bankruptcy itself.
I would firstly suggest seeing a debt counsellor at your local CAB before following this serious route, if it can be avoided at all, perhaps other arrangements can be made when all your facts are presented to the CAB. ?
It would be in your interests to see the CAB at your very earliest opportunity, before advising any creditors that you are intending taking the above serious step.