While local insurance brokers do, as Ethel suggests, offer the best chance of your family member getting insurance, I still wouldn't hold out too much hope.
Several years ago, I tried to help a friend get insurance on his house and contents. He had a very minor (non-custodial) conviction, which wasn't for any form of dishonesty. Between us we phoned nearly 100 brokers and insurance companies but not one would offer him cover. He eventually decided to wait until his conviction was spent before applying again. He moved away shortly afterwards and I've lost touch with him, so I don't know if he ever got cover. (The problem he faced was that, even after his conviction was 'spent', he would still have to answer 'Yes' to the question which asks 'Have you ever been declined cover?'. It's possible that he still might not be able to get insurance).
Although you say that your family member isn't a bank robber or rapist, it's worth remembering that, as far as the insurance companies are concerned, fraud is probably far worse than rape. (i.e. they're only concerned about minimising the risks of fraudulent claims. A sex offender might be totally honest when it comes to such matters but your family member has shown that he's prepared to consider fraudulent activity, so he presents a far greater risk).
It's probably fortunate that the conviction hasn't come to the attention of his bank. All UK High Street banks refuse account facilities to anyone with a conviction for fraud.
However, having typed out all of that rather negative response, I might have some good news for you. A bit of web-searching found this:
http://www.adrianflux.co.uk/fairplay-detail-go .php
Chris