The Chair of a parish or town council has virtually no powers as an individual and the Chair of a committee or sub-committee has even fewer. The role of Chair is to chair the meetings.
A well-run council reinvents itself at the Annual Meeting in May every year, with a complete review of every aspect of its operations. The framework for this is set out in para 5(j) of the model standing orders, which lists 21 items for consideration. Amongst them are a review of delegation arrangements to committees and staff, terms of reference for committees, membership of committees, creation of new committees etc.
Standing order 4(d) requires the Council to determine the number of meetings for each committee for the municipal year, set dates and times, agree venues, appoint the Chair etc. I take this into the Annual Meeting for clarity.
So the answer to your question is that it doesn't really matter whether or not the committee has a chair appointed by the Council, although it is bad practice not to. At the start of every committee meeting, those present decide who they want as Chair. However, the Clerk should be aware that the Council has not made a valid appointment and should add this to a Council agenda in order to be fully compliant.