Csdrhrt

When listening to Gander/Shannon ATC on shortwave, you could hear Concorde communicate its planned flight levels at longitudes from 20 West till 50 West. What I remember is that it would continue climbing to, say, flight level 570 at 30 West and then descent. So, it seems that it never reached cruising altitude, it would climb until halfway on the Atlantic and then start to descent again.

The simple answer is that the Concorde had no single assigned altitude, it was allowed to climb freely above ~FL450; this is discussed in depth in episode 166 – Flying the Concorde (worth the listen as it answers just about every Concorde question!). As @pilothead alludes to in their answer it climbed as it burned fuel but the aircraft never actually initiated a climb, it simply drifted up as it burned fuel and became lighter.

Also discussed in the episode is the complex approach and departure procedure. Due to the fuel burn schedule the Concorde did not really have the ability to hold for more than a single lap in a holding pattern or do a stepped climb with some route adjustments like many airliners. They had a special departure procedure that was more or less runway to cruise with no interruptions and a similar descent option. So the flight was effectively a climb to cruise block then a glide down right to landing. Depending on the wind and conditions of any given day as well as the load on board, the cruise altitude could vary greatly.

Concorde had a 10,000fpm climb and a max altitude of 60,000ft, so time to climb was not a problem. It had an optimum cruise altitude that varied with weight, so as it burned fuel it climbed higher to stay on the optimum.

There were no other aircraft operating at those altitudes, so it would get clearances to climb 15,000ft at a time and would cruise climb throughout the trip until descent to destination was required.

Amazing how we humans skew altitude and distance. 60,000 feet up is 12 miles. Transatlantic 2400 miles.
The climb would be a gradient of 12/1200 x 100 = 1.0%

I daresay the Concorde could climb a bit faster.
A 1.0% gradient would be barely noticeable in an automobile.