Stayed in London a few weeks ago. My recommendation is the Columbia Hotel at Lancaster Gate. Price is approx. 70 GBP for single - 89 GBP for double. This includes full English breakfast. Very central and very, very clean. Room included private bathroom. We booked a shuttle from Heathrow to hotel through a service called Dot2Dot. Cost was 17 GBP.
To get to Southampton, you can taxi to Victoria Coach Station (about 10GBP) and take a coach (National Express) to Southampton. A good place to stay in Southampton is the Dolphin Hotel. Very near the port. It is reasonable and again ... clean.