Yes, most rental companies don't do complete background or fiancial investigations. Immigration would require some money in the bank though. If bankrupt in the UK, I would personally try to make something work or resolve it as jobs in Cyprus are not easy to come by that pay enough to live well at the moment. Working hand to mouth in a new place and/or trying to escape problems at home doesn't usually turn out for the best especially if finances are limited. But, ultimately the answer to your question is yes, they could live here.