Relations between the British government and the Church of England, strained by the church's strong stand during the coal miners' strike, could undergo another test with the coming appointments of bishops.

Tension between church and state could get worse depending on the choices of the Crown Appointments Commission to fill at least five diocesan bishoprics. Unless the choices are free of political interference, the gulf between the spiritual and secular realms could grow.

The sees to be filled include some of the most prestigious in the church: Winchester, Bristol, Chelmsford, Exeter and Southwell.

Prime ministers usually accepted the commission's nominations, but some have reserved the right to ask the commission to think again, to give reasons for its choice, and even to advise the crown independently.

It is at this point that the five episcopal vacancies here take on a new significance. The government of Prime Minister Margaret Thatcher has been embarrassed and surprised at the way bishops, even Archbishop of Canterbury Robert Runcie himself, questioned the basis of national economic policy during the coal-mining crisis.

The search these days is for men who, along with wearing the episcopal finery, also possess something of the common touch. That may also include a political belief or two, and there may lie the quandary for Thatcher.