St Pancras in the rush hour isn't so bad. It's not the elaborately choreographed commuter ballet of Shinjuku, but it's certainly not a crush like Waterloo or Victoria.
The thing to remember about the new St Pancras is that it's effectively 4 mainline stations, with two Tube stations attached.
So, from Paddington, if you get the Tube, you need to get it from the Tube station on platform 15/16, not the one in the main building. Head up to the overbridge and follow signs to the "Hammersmith & City Line".
At King's Cross St Pancras, you'll arrive at the subsurface Underground station. Follow the signs for St Pancras, and you'll emerge at a lower level. Head up to the main concourse, and you'll emerge into what is basically an upmarket shopping mall in the undercroft of a stunning building.
You need the SouthEastern platforms, which are at the far end of the station, in the modern building. Keep left, head along to the end of the undercroft, then bear right (this is the only annoying bit, as you'll be crossing over the flow of people heading to/from the Thameslink section). You then have an escalator or elevators heading up to the SouthEastern platforms, past the big Starbucks. (If you want a coffee, try the Sourced Market place opposite. Their deli stuff is horrendously overpriced, but they do serve Monmouth Coffee, which is lovely.)
Trains will be busy, but you'll get a seat and luggage storage. Walk forward - people are lazy and the backs of trains fill up. If it looks particularly busy, it's only 15 minutes to the next train. Note that not all trains go to Ashford. Platforms tend to be announced 10-15 minutes before departure to manage crowds, so there can be a bit of a mad rush when they're posted.