Family NYE in Sydney

A good site to use is the NSW Transport Info site which shows you possible public transport availability / routes - site is https://transportnsw.info/#/
There is a trip planner on this site you can enter a street address and where you want to go to to see availability / travel times. As Jingili rightly points out Lane Cove is a sprawling suburb but is relatively close to the city; railway does run through the adjacent suburbs (St Leonards and Artarmon) but these stations not necessarily close to the swap house.
On another note there is a great shopping mall / centre in Lane Cove; a leafy suburb generally - nice place - also has a aquatic centre (public pool).

If you get a house SIT , not a swap/swop, there'll always be taxis though might be hard to get one NYE/D so book one ahead and then depending on where in Lane Cove you are, train stations may not be so far away, there being Artarmon, Chatswood and Rodeville Stations with the latter two pretty close to the Pacific Highway which is to the east of Lane Cove.
Weather is often quite balmy to warm and you'll likely find that you will not be alone in doing some hoofing of it.

Just plug your address into Google Maps and you'll see where you are.

I suggest that you plan to view the fireworks from whichever side of the harbour you are staying on.

If you are staying in the East, South or West of the city, then I would recommend Barangaroo Reserve. It's a ticketed spot ($30 each last year) that has limited numbers so is not overcrowded. While you can't see the Opera House, you have great views of the Bridge. https://www.sydneynewyearseve.com/vantage-points/barangaroo/ Sign up for their newsletter to be alerted when tickets go on sale.

If you are in the north, then Taronga Zoo is good, as are the National Park vantage spots on that side of the harbour. Places like Bradley's Head. https://www.nationalparks.nsw.gov.au/things-to-do/events/nye-in-sydney-harbour

I have some really detailed NYE posts on my blog if you would like to check them out. It's linked in my profile. Not sure if it's ok to include it here.

there'll always be taxis though might be hard to get one NYE/D so book one ahead and then depending on where in Lane Cove you are, train stations may not be so far away, there being Artarmon, Chatswood and Roseville Stations

It's worth emphasising that Sydney is a big city, and most suburbs cover a large area.

For example, from the nominal centre of Lane Cove to Chatswood Railway Station is more than 4 km, and depending on where the house is, it could be almost twice that. And this is not flat countryside either.

Your chances of obtaining taxis home - even with bookings - would be very slim indeed. Not trying to discount Lane Cove at all - you could do far worse - but just want to give you a sense of the scale.

Good points from above, recommending you stay on "your" side of the harbour (unless you do the cruise of course).

As I said KEAPPS, plot your possible whereabouts on a map and in having a look at one of the Sydney Area, you will be able to see that there is Lane Cove National Park with urban housing between there and the Pacific Highway/Railway line.
The National Park itself is about 3 km. in length with the Lane Cove river running through it and it is approximately 2 km. from the eastern side of the NP across to the Pacific Highway and then there is some additional parklands a bit further to the north west, also with the Lane Cove river ( or more just a trickle there ) running through and some people may refer to that area as Lane Cove too but also other suburban names and there is another railway line which runs up towards that parkland area, Epping and Cheltenham stations but a tad further away.

But as I said, have a look where you might be and you can gauge what sort of a walk you might have, maybe 2 -3 km. at most and quite possibly a lot less.

and there is another railway line which runs up towards that parkland area, Epping and Cheltenham stations but a tad further away.

It's a long tad ... Cheltenham Station is 13 km from the nominal centre of Lane Cove, with Epping nearly as far.

I lived in Sydney for 25 years, including quite close to Lane Cove on both east and west sides, and do have a pretty good sense of the scale. The bus services will be your best option, in just about every way.

KEAPPS, just plot where you might be and see what is available and as one who has also lived in Sydney and visits quite regularly ( if not for NYE ), I am not suggesting that you walk too far at all.

You may even find buses OK if the railway lines do not get you close enough but then none of us know where you might be.

I grew up in Lane Cove and also lived there as an adult. There are many frequent buses - and depending on where in the suburb you can get into the city in 20 minutes. From some spots in lane cove you can see the harbour and the harbour bridge fireworks - it could be a good option if the adults still wanted to see the midnight fireworks but the children only the family ones

The park at the end of Glebe Point Road is also a good spot to view the fireworks. I've done it a couple of times (although admittedly not for a few years now) and it's never been packed out like the vantage points to the east of the bridge. Balmain is also a good option, although I think they stop the ferry service in the early evening to clear the harbour ahead of the fireworks.

Do you have any contacts in Sydney KEAPPS? If you do, it might be worth asking them to put the word out through their networks to see if anyone is leaving town over the Christmas/New Year period and might be interested in having someone house-sit while they're away.

Yes ... we have friends with an apartment quite close to the park at the end of Glebe Point Road, and have been there a couple of times too (not for a decade though). It was certainly well attended by "locals", with a good picnic atmosphere, but not overly crowded at all.

You have a view of the "rear" side of the Harbour Bridge, and it's not bad ... not quite the same as from the "front" side of Sydney Cove and the Opera House, but still not bad at all. Although I don't know whether any new buildings at Barangaroo are now obstructing the view.

It could be rather complicated to get from there to Lane Cove however, on New Year's Eve, I expect.