TL;DR: Waterproof boots and yaktrax/microspikes/etc are a good idea.
I've been craving some time in the mountains, a little bit of snow but not too much, but wasn't eager to hike in the rain. Weather cleared up and after seeing a post in a hiking group about Talapus Lake, decided it would make a nice sunny winter hike with the kids. We did not continue to Olallie Lake so I can't comment on trail conditions after Talapus.
We arrived at 9:30 am and the parking lot had 7 or 8 cars. By the time we returned (~2 pm), the parking lot was full and several cars were parked alongside the road. There is a bathroom but no TP. While the road up was snow-free (and ice-free except the last little bit), the parking lot had deep, slippery slush. The area around the toilet and first section of trail are very wet, with running and standing water and a lot of mud. It clears up before you lose sight of the parking lot though, and most of the trail is good dirt trail until you hit the snow line.
It was windy for us, so colder feeling than I expected for a 45+ degree day. Despite last week's weather, the trail and road in are completely clear of blowdowns and branches. While its uphill most of the way to the lake, the grade is gentle. Our 3-year old* was able to walk himself until we hit the slick area. The first icy patch was short and slushy but it got progressively more slippery after that so we put on yaktrax and continued. You might like poles, especially closer to the lake, but we were fine without. Less than a mile before the lake, the snow gets deeper and we were basically walking in a single track in compact snow.
There are some bridge crossings where the snow is mounded over the bridge, but you'll be fine if you have at least decent traction on your boots (my dad and little brother didn't have extra traction on their boots and did okay on most of it). The one exception is a narrow log crossing that has 6"+ of compact snow on top. I carried the toddler over and we were okay, but I wouldn't recommend letting kids cross themselves, or going over without traction. There is a path down and over the creek using rocks that is less slippery if you're okay getting closer to the water.
The snow-covered forest is gorgeous right now, and the lake is frozen and snowy (although looked a little melty towards the middle). We followed a little trail down to a flat spot on the shore, and had lunch while taking in the clean air. The birds were hungry (and fat), and SUPER aggressive, please don't feed them. They were practically dive-bombing us, trying to knock food out of our hands and pick up any dropped pieces. A few small snowballs shooed them away for a bit.
The walk back was easy and mostly downhill. A good number of people on the trail and everyone looked happy and well-prepared for conditions.
*A note, if you are hiking with younger kids:
This looks to be a family friendly hike during the summer but it is very cold and slippery after the first 1.5 mi or so. It was windy, wet, and they don't make yaktrax/etc for kids, so it hard to keep them stable. If you don't have the cold-weather gear (waterproof and warm layers, gloves, hats, traction/carrier + traction for yourself), I would save this one for late spring/summer.