Reddit mentions of Laing LHB08100092 AutoCirc Recirculation Pump with Timer

1. Do other parts of your washing routing first, like brushing your teeth, contact lenses, etc. that are less sensitive to cold water. Do this at a sink nearby the shower, and use only "hot" water. This will use up at least some of the cold water from the hot water pipe for something useful. Bonus: you can put a small basin in your sink to capture the water from when you are washing up, and empty that into the buckets in your tub/shower for flushing.
   
2. When warming up the water in the shower, set it to maximum hot so only water from the hot water pipe is being flushed and not also water from the cold water pipe. Put a bucket to capture this water and then use it for flushing toilets.
   
3. Put as many buckets or basins as you can stand on the floor of the shower while you are showering. Turn on the water to get wet, then turn it off to scrub, then turn it back on to rinse.
   
4. Install a hot water recirculation pump, which will pump the cold water from the hot water pipe into the cold water pipe (yes, it does create some warm water in the cold pipe for a little bit) when the hot pipe temp drops below a set threshold. I installed one of these under the sink closest to the shower farthest away from the water heater and connected it to a smart switch so I can tell Alexa to "turn on hot water recirculation" a few minutes before I shower.

So I got this:

https://www.amazon.com/gp/aw/d/B0046MDAK0/ref=ya_aw_od_piI

We have a tankless hot water, it takes some time for the hot to make it, so I figure this will kill two birds. 1 hot water is closer to the items, 2 water is moving through pipes in winter.

Have tankless though so not sure what will happen. I did set it to the lowest amount of time turned on, the same time off, and repeat. I can explain more if necessary. The idea was in case it's not enough water to actually trigger the hot water from the heater.

I put one of these in and it has a thermostat inside so it only runs long enough to get hot water to the pump then shuts off until the water starts to cool down. It also has a timer so you set it to only run during the day if you like. It does not really run that much and I noticed no change in my electric usage.

Correct. I installed one of these 20 years ago and it's awesome. When you turn on a hot water faucet and wait for it to warm up the water that you're wasting down the drain used to be hot. The pump is thermostatically controlled and silently pulls water as needed. It sends the water back to the water heater through the cold water line so no water is wasted.

This may not work for OP if an oil fired system literally doesn't keep a tank of hot water ready at all times. It's also probably not a good idea if your pipes run through a concrete slab as that would waste a lot of heat.