Assuming your wife expects to drive on the Bakerloo line (ie the brown line), then at this time of day it is unlikely that the trains will be full at busy times, as the celebrations start earlier and end later and are centered at considerable distance West and on the Central Line and Hammersmith & City Line. There are likely to be more people changing the subway lines than on an average Monday lunchtime, especially since the change from one subway line to the other in Paddington may be linked to a walk through the central station , Once in London, it will feel much more like a Saturday than a weekday.
The cheaper option is to use the subway for the entire journey and change from the Piccadilly line to Bakerloo in Piccadilly Circus. However, this is likely to feel significantly crowded / depressing than the Heathrow Express, and the Piccadilly Circus will take much longer is never a particularly quiet station, especially over a national holiday.
The most expensive is to take a taxi from Heathrow or Paddington. If you are using a Heathrow taxi, there are road closures because of the carnival, but I do not think they are interfering with some of the more obvious routes, and your taxi driver should be aware of this. These two options would also eliminate most of the difficulty with steps and luggage that occur when using the tube. Nevertheless, it is not a particularly pleasant or beautiful trip from any point involving either a freeway or the second most polluted road in London.