The covid swab we recommend getting first 5 days after symptoms, false negatives can happen. In the context of wife having covid I would say both have covid.
Continous cough is to be expected, so is difficulty breathing but the later is more worrying (especially if any chest pain is involved).
Main issue is how severe the breathing difficulty is, whether its got them doing less activities and how much, is the breathlessness exertional or at rest? Best thing would be to ring the GP or 111 as its Sunday and they can both have vitals checked at a covid hot hub.
If symptoms persist then both can be referred to long covid clinic as well now but might depend on local availability (most places have them)