Yes, this was part of the procedure for a "crash dive." It was standard practice for submarines at the time, which spent most of their time on the surface and occasionally had to submerge on short notice to avoid being spotted or attacked by enemies.

I can't seem to place a specific source that explicitly describes the crew moving forward, but it was common and practiced by German U-boats and other navies at the time. This translation of a 1943 publication on "Diving Regulations for U-boats" describes the technical procedure of what you're seeing in the movie, starting on page 34, line 111. Crash dives were supposed to be practiced over and over again, including in specific conditions, to ensure the crew was ready when it had to make one for real.

The crash dive is a wartime maneuver and forms the basis of front training of a crew. Therefore, routine preliminary training for crash diving including timing is particularly important. It is to be done as often as possible and gradually to be made more demanding. Only the certainty that every man of the crew handles the measures and controls he is responsible for with absolute certainty allows the Commander to carry out the crash dive in wartime.

Later in the same section, starting at 115:

Order: "Alarm!"

The helmsman sounds the alarm bell and shouts "Alarm". He immediately puts both engine telegraphs on "Dive", switches the receivers to "Both rooms", turns off the alarm bell and, finally, switches the engine telegraph receivers to "E motor". The engine telegraphs remain set on "Dive", until the E motor is set for Ahead GF or another propulsion level is ordered by the Commander.

Every man of the crew repeats the order aloud, so that it is guaranteed that even the sleepers are awakened by the order. Particular attention must be paid to the loud exclamation of the order because the noise still prevalent in the boat must be overcome and the order must get into the diesel engine room even in the event of the failure of the alarm bell, flashing light and engine telegraphs.

As you see in the movie, not every sailor was supposed to run forward; some had to stay at specific duty stations during the dive. But even during normal dives (not crash dives), crew were supposed to be in specific places to maintain the proper "trim" on the boat. From page 25, line 84:

If trimming with people is necessary, this should be carried out within the fore- and aftership, e.g., "Bow room two men forward". On the order:

Engineering officer: "Rooms forward" respectively "astern".

The crew is to run to the front or back of their compartments to the forward or aft watertight bulkhead. On the order:

Engineering officer: "Section forward" respectively "astern".

They run in each case up to the pressure resistant bulkhead in the control room or in the end rooms.

If particularly strong trim effects are required, it is extended to the entire length of the boat:

Engineering officer: "All men forward or astern".

In each case only dispensable sailors run. Helmsmen, control room teams and the operating team of the motors remain at their posts.

At the order:

Engineering Officer: "To stations".

The sailors run again to their diving stations according to their role.

The time sequence for a crash dive in the movie is also realistic; it shows the whole process taking about a minute during training. Well-trained crews could do it even faster. According to this US Navy report, some crews could do it in half that time.

Given fair visibility, lookouts can spot an approaching aircraft 6 miles away, which gives them time to crash dive. (Another prisoner stated that U-boat crews were trained to submerge within 30 seconds after sighting an aircraft. He thought that the record for crash diving, starting with four men on the bridge, was 28 seconds.)