(i) But, in the sense of except is a preposition and takes the objective case. So it is not ungrammatical to say "Nobody but him attended the meeting". But it is more usual to treat but in that position as a conjunction and say: "Nobody but he attended the meeting".
(ii) In using but as a conjunction an easy slip is to put it where there should be an and, forgetting that the conjunction that you want is one that does not go contrary to the clause immediately preceding but continues in the same sense.
It is agreed that the primary condition of the scheme is satisfied, but it is also necessary to establish that your war service interrupted an organised course of study for a professional qualification comparable to that for which application is made, but as explained in previous letters you are unable to fulfil this condition.
The italicised but should be and. The line of thought has already been turned by the first but; it is now going straight on.
A similar slip is made in:
The Forestry Commission will probably only be able to offer you a post as a Forest Labourer, or possibly in leading a gang of forest workers, but there are at the moment no vacancies for Forest Officers.
Either only must be omitted or the but must be changed to since.
(iii) There is no ground for the idea that it is incorrect to begin a sentence with but or and. See And