Distance: 5.9 km (3.0 DOC hours) - Marked route - Easy-moderate terrain
Altitude: 1182m to 1866m. Gain: 1075m. Loss: 391m . Gradient: 15 deg (Moderate-hard)
Skills: Occasional scrambles (3/7) - Streams (2/6)

Marked route (6/9)
Uncut route, with markers normally within sight

Primary, mapped (1/5)
Formal route, on maps, principal access to hut or along range / catchment

Moderate-hard (difficulty: 4/5)
Continuous moderate climbs

Easy-moderate (difficulty: 2/5)
Tussock / river-gravel / open bush / cut track

Occasional scrambles (difficulty: 3/7)
Tramp with some scrambles and exposure to falls

Streams (difficulty: 2/6)
Small streams, danger only in extreme weather

Unknown (difficulty: 0/7)
Not specified

From Blue Lake, the route is poled all the way to Waiau Pass (and on to the valley floor).
Initially gently rising and open, once to the trees the track steepens. A rising track leads to the true left of Lake Constance brings you to a steep scree climb then a traverse to the mid-point of the lake.
Although there have been recorded deaths above the lake, the track is straight forward. However, take care if you choose to approach the top of the bluffs as the fall to the lake is sheer.
The poled route leads to a steep descent in a gut and to the lake shore – then along to the lake head and across the valley floor (good campsites 100m before the start of the climb) and up up up to Waiau Pass.
It’s steep but the initial section on larger rocks is stable. The track moves onto a grass tongue until an obvious footpad angles up, across the scree, towards a grassy shelf. A useful tarn at the base of the final ascent to the pass means less water needs to be hauled up from the valley.

Note: Described in the reverse direction to your journey

Descending Waiau Pass on the North side into the Lake Constance basin is a straightforward descent in scree, following a poled route. The route crosses the stream at the head of the valley, followed Lake Constance on the left (West) (Figure 21a), climbs above bluffs at the lake head, over a large boulder field, after which views of the beautiful Blue Lake / Rotomariewhenua are gained (Figure 21b).

Distance: 1.6 km (1.0 DOC hours) - Marked route
Altitude: 1300m to 1868m. Gain: 24m. Loss: 590m . Gradient: 23 deg (Moderate-hard)
Skills: Prolonged scrambles (4/7)

Marked route (6/9)
Uncut route, with markers normally within sight

Primary, unmapped (2/5)
Route not on maps, but is principal access to hut or along range / catchment

Moderate-hard (difficulty: 4/5)
Continuous moderate climbs

Unknown (difficulty: 0/5)
Not specified

Prolonged scrambles (difficulty: 4/7)
Scramble, frequently on exposed ridges, spurs, faces

None (difficulty: 1/6)
No unbridged river crossings

Unknown (difficulty: 0/7)
Not specified

On the map, this route looks steep and risky - in wet or icy weather it would be. However, poled all the way, in good weather this is a simple descent on good, grippy rock.

Note: Described in the reverse direction to your journey

@millera wrote: Being a poled route, Waiau Pass is reasonably straight forward, with some impressive rocky climbs, requiring stowing of walking poles and clambering up rocks with hands and feet. It flattens significantly near the top, with an easy route in scree (Figure 20a), and views back to Lake Thompson (Figure 20b).

Distance: 2.7 km (2.5 DOC hours) - Unmarked route, clear - Moderate terrain
Altitude: 1306m to 1791m. Gain: 510m. Loss: 29m . Gradient: 12 deg (Moderate-hard)
Skills: Occasional scrambles (3/7) - Streams (2/6)

Unmarked route, clear (7/9)
Unmarked route, follows obvious geography (river, spur, trail, etc)

Primary, unmapped (2/5)
Route not on maps, but is principal access to hut or along range / catchment

Moderate-hard (difficulty: 4/5)
Continuous moderate climbs

Moderate (difficulty: 3/5)
Low scrub in open or bush / river rocks

Occasional scrambles (difficulty: 3/7)
Tramp with some scrambles and exposure to falls

Streams (difficulty: 2/6)
Small streams, danger only in extreme weather

Unknown (difficulty: 0/7)
Not specified

Crossing the river to the TR, we tried to cut the corner but travel was easier closer to the stream.

Crossing back to the TL involved a scramble down and up out of the stream. Then a stiff climb to the gut (and fault line) that runs up to Lake Thompson. We traveled in the gut but travel would also be good on the ridgetop. From Lake Thompson, it's a short climb to Thompson Pass, first on a grassy tongue, then on rock.

Note: Described in the reverse direction to your journey

(notes from duplicate route by @millera)

Lake Thompson is a majestic setting and a lovely place to camp in good weather. I would describe it as slightly smaller than Lake Angelus, in a similar majestic mountain setting, but without the hut and people.

The descent from Lake Thompson begins down the obvious earthquake scarp to the west of the lake (Figure 18a) – avoid exiting from the main stream draining the lake, as this leads to bluffs and waterfalls. The descent became a narrow valley, which then became steep with vertical drops after turning east (Figure 18b). I made for the ridge to the true left (Figure 19a) which gave easy walking all the way down, re-joining the river close to the Waiau Pass turnoff. This is the beginning of the marked route, part of the Waiau Pass route and the Te Araroa Trail (Figure 19b).

Distance: 7.6 km (3.0 DOC hours) - Unmarked route, clear - Moderate terrain
Altitude: 896m to 1247m. Gain: 172m. Loss: 523m . Gradient: 5 deg (Moderate)
Skills: - Prolonged rivers (4/6) Winter - Snow/ice underfoot, avalanche risk (3/7)

Unmarked route, clear (7/9)
Unmarked route, follows obvious geography (river, spur, trail, etc)

Primary, unmapped (2/5)
Route not on maps, but is principal access to hut or along range / catchment

Moderate (difficulty: 3/5)
Occasional moderate climbs

Moderate (difficulty: 3/5)
Low scrub in open or bush / river rocks

None (difficulty: 1/7)
No alpine skills required

Prolonged rivers (difficulty: 4/6)
Frequent river travel - impassible after rain, risk of getting trapped

Snow/ice underfoot, avalanche risk (difficulty: 3/7)
Snow/ice likely on track, tramp crosses avalance paths

From the point where the Davids Saddle route joins the Matakitaki valley downstream, there are intermittent footpads, requiring frequent river crossings to pick up easier travel on the grassy flats. Just 2km from the hut, on the TL, is a benched cut 1km track, initially climbing away from the river and then back to the flats. 1km above the hut, we crossed to the grassy flat on the TR, crossing back to the TL at the hut.

Note: Described in the reverse direction to your journey

(updated with notes from duplicate route by Millera)

This was the most uncertain part of the trip. Hence I have included a lot of detail for those considering it. The head of the East Matakitaki is a beautiful valley, and would make a lovely place to camp.

No track is marked on the 1:50,000 topo map upstream from the East Matakitaki Hut. However, travel for approximately 1km upstream from the hut is easy going on the true left on river flats, with two river crossings deemed preferable to avoid a steep bush clad bank on the true left. At the river flow, with no significant rain for some days, these were thigh deep in places but easily manageable.
At the end of the river flats the slopes either side of the river become steep, with what appeared to be a gorge-like section. From here, at 930 m, a track has been made through the forest on the true left for about 1km, despite the map showing none. It begins roughly 70 m away from the river on the true left, at the bush edge. It is worth finding this, as the travel is reasonably easy going on this marked track, with markers reasonably frequent.

Beyond the end of this track there are sections of boulders, re-entering the bush, and unmarked track. From the clearing at 990 m travel is possible on the river flats on the true left. After this clearing the track re-enters bush on the true left, where it is rough and very overgrown, but avoids travelling in the riverbed / river crossings. The track then emerges to another clearing, crossing a side-stream from the true left at about 1035 m.