tl;dr: An observer on the "Moon side" would see only half the phases during the fortnight-long night: from waxing half-moon to waning half-moon. They would also see the Moon during the day (early morning, late afternoon), as we currently can, but they would never see the New Moon or crescent moons at night since these would be only visible during the day.
While the Earth being tidally locked to the Moon would indeed mean that the side of the Earth facing the Moon always sees the Moon, the phases of the Moon result from the angle between the Earth, the Moon and the Sun, and probably wouldn’t change significantly.
The Moon orbits around the Earth-Moon barycentre with an orbital period of 27.3 days and a synodic period (i.e. from Full Moon to Full Moon) of 29.5 days. An observer on the Moon-facing side of the Earth would certainly see the Moon go through its different phases, but it would still take 29.5 days. [For the purposes of this question I'm ignoring the increasing lunar period due to the Moon receding from the Earth.]
As the question notes, one obvious outcome of Earth becoming tidally locked to the Moon is that Earth’s “day” becomes the same as the Moon’s synodic period: 29.5 days. Daylight will last for about 15 days, baking everything other than at the poles. The 360-hour night will be freezing, even in the tropics. Local weather will be utterly different.
Another notable outcome is there will only be really tiny “solar” tides spaced about 15 days apart (due to the Earth’s slow rotation in relation to the Sun). The loss of significant tidal flows would probably have a dramatic negative effect on marine life.
The Earth tidally locked to the Moon is a nightmare scenario!
The core question
So standing on the "lucky" side of the Earth you should be able to see
all the lunar phases during a single night. The Moon become gradually
more illuminated, peak as a full Moon and then darken as the night
No: you would only see only half the phases during the fortnight-long night: from waxing half-moon to waning half-moon. You would also see the Moon during the day (early morning, late afternoon), as we currently can, but you would never see the New Moon or crescent moons at night since these would be only visible during the day.
We currently see the New Moon because when it's roughly overhead for some at midday, it's not far above the horizon just before sunrise or just after sunset for others. Since the New Moon phase currently lasts about 3-5 days (depending on how you define "new"), but the Earth rotates in just one 24-hour day, most habitable places on Earth get to see this phase every lunar month. However, in the tidally-locked-Earth scenario, the phase still lasts 3-5 days but the Earth rotates once every 29.5 days. Those viewing the Moon just after sunset will only ever see the Moon in that position above the horizon – it more or less "sits" there, apart from libration and nodal period effects – and they will miss some of the other phases because the Sun will be too bright at, say, the waning half-moon.
EDIT - Two additional things to note:
(1) Earth's phases
While the Earth becoming tidally locked to the Moon is hypothetical, the Moon is already tidally locked to the Earth, which means the above descriptions can be applied in reverse for a real observer on the Moon:
- a lunar "day" (sunrise to sunrise) is 29.5 Earth days;
- the Earth's position in the lunar sky is relatively constant over the lunar day, within about 15º (due to a slow but noticeable libration);
- the Moon's orbit is elliptical, so over much longer periods the Earth would alternately recede and grow larger (a "Super Earth");
- unlike the Earth-locked scenario, all of the Earth's phases – from Full Earth at lunar midnight to New Earth at lunar noon - would be visible, because there's no atmosphere and hence no "bright daytime sky" to block out the Earthlight.
Note also that the famous Earthrise experience of the Apollo 8 astronauts would not be available to an observer on the Moon's surface. The Earth "rose" above the limb of the Moon because the astronauts were in orbit. Nonetheless, someone located on the border of the "dark side of the Moon" would see the Earth on the lunar horizon, and libration could result in the Earth alternately setting below the horizon and rising again over a lunar day. For that observer, however, earthrise and earthset occur at roughly the same point on the horizon!
(2) Will the Earth really end up tidally locked to the Moon?
Tidal braking is slowing the Earth's rotation by 23 milliseconds (and increasing the Moon's distance from us by 38 metres) per 1,000 years. At that rate, the Andromeda Galaxy will have collided with the Milky Way long before the Earth's rotation has slowed down enough to approach tidal locking.
In fact, this AstronomySE answer suggests that as the Earth's oceans "are responsible for almost all of the tidal deceleration" (David Hammen's comment), the evaporation of the oceans in a billion years will eliminate the main drag on the Earth's spin and tidal locking may therefore never occur.