As long as a tenant keeps to all other aspects of the agreement, they have a fundamental legal right to stay for 6 months. Consequently, none of the offical notices from the landlord to gain possession would be valid during this period. There is no reciprocal right for the landlord to force the tenant to stay for the entire period. For this reason, to make it fair for both parties, it is normal for both parties to agree to an initial 6 month fixed term tenancy.
The council is wrong. The tenancy is not illegal, the 3 month length is unenforcible. The normal rules on illegal eviction apply if any type of force is used by the landlord, presumably the £5,000 fine and imprisonment they mention.
If they didn't pay rent, they could have evicted them one or more of grounds 8, 10 or 11. See -
http://www.statutelaw.gov.uk/content.aspx?LegType=All+Primary&PageNumber=39&NavFrom=2&parentActiveTextDocId=2128236&ActiveTextDocId=2128476&filesize=4438