Hi Richard
Wow you could have some monster swede. They should just carry on growing if there is enough room and as long as moisture is present. They do like damp soil so leaves dying off is more to do with drought than maturity, and I expect they will resume healthy growth when it rains again, right up to Christmas if it stays mild.
I sowed swedes here in early June in modules and planted them out after clearing some early salads, so they are still small. Whereas my parsnips, sown in March, are already pretty large and could be eaten (but like you I don’t want to, so they can grow some more). The beauty of these winter vegetables is that they can stay in the ground a long time until you need them.