There are lots of issues to unpick in this question. Let's start with the easiest point.
All schools should have a contingency plan for someone to take over when the headteacher is out of school. This is usually the deputy head. Schools do not have to employ a deputy, but they must have a senior manager whose job description includes standing in for the head at short notice.
This covers short-term absence, but I suspect that this issue concerns a long-term illness. In these cases the governing body will have to formalise the arrangement, and appoint someone as an acting headteacher. Normally this will be the deputy or senior manager, but sometimes that individual does not want the additional responsibility - and work. In these cases the governing body may need to seek a temporary acting head from outside the school.
The local authority should be able to help here, but a simple and effective solution is to "borrow" a capable deputy from another school. This has the double advantage of solving the immediate problem while offering some valuable headship experience to a deputy head.
This does involve additional costs: the partner school has lost a capable member of staff, albeit temporarily, and your school will need to compensate them for that loss. You will also need to reward the acting head. He or she can continue to draw a salary from the partner school, but they will be doing the work of a head, so they will need to be paid accordingly.
But there is another issue tied up in this question, which seems to assume that the head makes all the strategic decisions in the school. This may be the reality in some schools where the governing bodies "rubber stamp" the head's decisions, but it is not what should happen - and it is not the law.
The governing body is the strategic decision-making body, which means that governors set the strategy. The head's role is to advise and to offer options for the governors, then to put into operation the strategy that the governing body has decided upon. So in this situation the acting head would advise the governing body.
The head may provide a strategic plan, but it is the governing body that approves it - or not. Governors may wish to see such a plan amended, or they may throw it out altogether. The head has the option to be a governor, so he or she can influence the decision and it would be a foolish governing body that did not take note of advice from the headteacher. But the governing body is the final decision maker.
Clare Collins is chairwoman of the National Governors' Association. www.nga.org.uk.