Refugee Convention, which is the statute under which a person can claim asylum who: "... owing to a well-founded fear of being persecuted for reasons of race, religion, nationality, membership of a particular social group or political opinion, is outside the country of his nationality and is unable or, owing to such fear, is unwilling to avail himself of the protection of that country; or who, not having a nationality and being outside the country of his former habitual residence as a result of such events, is unable or, owing to such fear, is unwilling to return to it.."
So, claiming asylum comes down to whether someone has a well-founded fear, i.e. good reasons to back up why they are afraid.
Current caselaw (RN Zimbabwe) confirms that Zimbabweans who have lived in the UK for a significant length of time can be viewed as being at an enhanced risk if they were to return to Zimbabwe, due to the strong body of support for the Movement for Democratic Change in the UK. So, there is likely to be a risk to them, but it all depends on the individuals concerned in terms of political involvement, family political allegiances in Zimbabwe, tribe (to an extent - Ndebele people suffered dreadfully under Mugabe in the 1980s).
Incidentally, the Home Office formerly had a policy whereby if a child had lived in the UK for seven years, that would be a strong factor in considering granting leave. However it was scrapped several years ago.
In short, whether or not this particular family have a case to stay or not comes down to their individual history and experience. It's not helpful to launch into the "illegal immigrants" rhetoric. These are people who were formerly legal. Now that leave has come to an end. If they stay with no status, they are overstayers. If they claim asylum, they are asylum seekers with temporary admission. They did not enter illegally nor, it seems on the face of it, attempt to deceive in order to extend leave.