Hmmm,

Let me answer the question as a Canadian who studied the US system many years ago and who is in the US now.

In the UK the executive (cabinet) and legislative branches are are in parliament.

In the US Congress does legislation and the President appoints the cabinet form outside congress.

If Congress becomes Republican then Obama will still run the cabinet departments but he will not be able to get new laws passed.

At this time the Republicans are threatening to refuse funding to laws that previous Congresses have passed and to refuse funding to support Supreme Court decisions with which they disagree. If they do this, things will be bad.