We will be arriving in early April into Gatwick at 6:30AM on a Tuesday, so we will be on the train by 7:30am to 8:00am. We will each have one large suitcase plus carry on luggage. Will either of the suggested routes from Gatwick airport to Paddington Station allow us to travel the underground without steps?
Neither of those routes are step-free - there's no step-free access onto the Circle/District at Victoria, nor is there step-free access from the Southern half of the Circle/District platforms at Paddington. However, Thameslink to Farringdon and changing onto the Circle or Hammersmith & City should be step-free. Farringdon, I believe, has step-free access between Thameslink and the Circle/Hammersmith & City platforms via lifts (though I think you might have to use two sets of lifts, which might be irritating), and Paddington Northern half of the Circle/Hammersmith & City has lifts.
(There's confusion at Paddington because the Circle line uses two completely different sets of platforms depending on which direction you're coming from - the platforms from the northern half of the circle line shared with the Hammersmith & City, as opposed to the southern half of the circle line shared with the District, are step-free.)
Another option which requires a change of trains, if you're OK with routes that involve escalators but not stairs, is to get the Victoria line from Victoria (step-free) and change at Oxford Circus (step-free for same-direction interchange) onto the Bakerloo line to Paddington (which has escalators but not lifts).
Source: http://content.tfl.gov.uk/step-free-tube-guide-map.pdf and http://content.tfl.gov.uk/avoiding-stairs-tube-guide.pdf
The Journey Planner site Travellines will search for step free routes if asked.
Entering Gatwick to Paddington with Step free gives as fastest.
Gatwick to Victoria by Southern train.
Then take TfL bus 36 to Paddington.
Note you are going to London at peak rush hour so you could find trains and busses full especially with luggage. (I would consider a taxi from Victoria to Paddington)
As a slightly (read very) left-field option, National Express coaches run from Gatwick to Heathrow Airport, and the Heathrow Express train service connects Heathrow to Paddington Station. This journey won't score highly in terms of speed or cost, but does avoid crossing central London with luggage, and makes maximal use of the fact that airports are well designed for people with luggage.
Alternatively, if you are making an onward journey from Paddington and it's long distance rather than suburban, Great Western Railways run services from Gatwick Airport to Reading Station, which is usually the first stop after Paddington for fast trains.
