Activate 7 day Travel card at beginning of trip or after three days?

How old are your children because under 11s are free and 11-18 year olds will need a photo Oyster card anyway if you want them to access child fares.

I am no fare expert but as Heathrow is in zone 6 then I would think it wouldn’t be good to start a week travelcard there as you’d need it be a zone 1-6 card = more expensive than a zone 1-2 card.

You can select the day you want the 7day travel card to start when you purchase the Oyster card and add PAYG (pay as you go funds).
If you're not taking the tube, then when you do purchase it, same thing. For kids 11-15, you can have the Young Visitor's Discount added and their fares will be half, I think.
Depending on how many days you will be out of London, it might be cheaper (but less convenient) to use PAYG instead of a travel card. You would just have to be aware of how much is being spent each day and top up if necessary. I would try to schedule the dates of the Travel card to coincide with the days of most travel in London.

York has tons to see, and I wouldn't attempt it as a day trip from London. However, if you must, consider that you're highly unlikely to take more than two London transit trips that day (hotel to train station and vice versa). Therefore, the York day should ideally be one of your first or one of your last days in London so it falls outside the weekly-travel-card validity period.

The weekly travel card may well save a bit of money (at least for adults--I don't know about the youngsters) if you schedule it around your out-of-London days. However, there's not a vast difference between the cost of the travel card and seven times the daily fare cap you get automatically with pay-as-you-go fares. Any day you don't take enough trips to bump into the cap reduces the relative advantage of the weekly travel card. For example, if you spend a day seeing Westminster Abbey and the Churchill War Rooms, just taking the Underground back and forth from your hotel, that's not enough to hit the cap.

You need to do the math for what you intend to do, but I suspect that Helen is probably right. Day 1 and Day 10 are one way trips to/ from LHR, and you probably won't want to do much other travelling on Day 1 with jet lag.
That leaves 8 days, 4 of which are side trips, three of which are out of town.
Windsor is beyond travelcard. If you can't do contactless for the kids that would need to be day return tickets anyway.
Do Richmond maybe as PAYG it's in Zone 4.
You might even want to take the river boat to Richmond one or both ways.
So for the other 4 days using PAYG daily price capping may turn out as the cheaper option,
It sort of depends on your plans, or if you just want the ease of a Z12 travelcard, and are happy that it might cost you a bit over the odds. The difference won't be that great- a few ££ one way or the other.

By the way for the day trip to Eastbourne the Southern Railway day save ticket is almost certainly the best value day ticket. Buy it on line at least 3 days ahead of travel. Check it out on line and see what you think.- https://www.nationalrail.co.uk/tickets-railcards-offers/promotions/prbd690bc35ce74100a7ae34c3df5fff/
Going to Windsor a Group Save ticket is probably going to be the best value fare. Something to look at.
To York there is a ticket called the LNER family return for £105 which may turn out to be the best value ticket, which is for 2 adults and between 1 and 4 children.
It all depends on how many children and their ages.
A bit to ponder about here.

Thanks to everyone who has shared their advice here. After reading through your thoughts and looking at our itinerary it may not make sense to do a 7 day travel card and just stick with the oyster PAYG. I think had we not wanted to include the out of London excursions, it would make more sense. You all explained the breakdown so well and I really appreciate it.

For example, we’re thinking of going to Eastbourne/ Seven Sisters area on Tuesday and York on Friday (we got early tickets for york to keep costs down- yay). It seems that York trip is not always seen as a day trip option from London, but it holds a special place in the heart of my daughter and we really wanted to go. It will be a long day, for sure, but hopefully a good one! (Also, my kids aren’t “kids” to anyone but me - they are in their mid- late 20s - sorry for the confusion on that 😊) isn31c, I’ll definitely use your link for the southeast trip- thank you!

We are all getting more excited as the trip approaches- especially seeing all the landmarks on the news with the coronation tomorrow! Enjoy your weekend everyone!

You will get weekly capping automatically for the period of your stay falling between Monday and Sunday, should you reach this.