The trains to Weymouth are operated by southwest trains who are not involved in any industrial action.
So your main options both involve a tube ride from St Pancras:
To Victoria for the coach station where you can get to Weymouth in about 5 hours changing at Bournemouth or Bristol ranging dependent on timing between £7-20 one way, there is only one direct service a day leaving London at 4pm (that's the £7 one), or
To Waterloo for the train, which runs direct taking just under 3 hours for about £14 if bought in advance (I'm looking here at prices payable today for travel in 2 weeks time)
Very helpful - many thanks - train looks best!
The only negative with southwest trains (who have not been on strike ever as far as I remember) is the lack of luggage space. They use commuter train carriages which don't have luggage space at the end. In the middle of the day half the seats will be empty so it's no problem. But at rush hour it is.
About half the train carriages have a bicycle symbol painted on the door. If there are no bicycles, you could potentially put large luggage in the bike storage here. But it's hard to keep an eye on it there (door closes between it and the carriage seating). Small bags can go overhead on the train, but if you have large luggage, especially if you cannot lift it, you might find the coach a better option.
After Southampton it's likely the train will be quiet (possibly it will even split. Listen to the announcement if they say for Weymouth you must be in the front 5 carriages). There is a trolley service selling refreshments (should come to you at your seat, but feel free to go and find it in case it's been blocked in by standing passengers or bags). You will see the New Forest National Park en route.
SWT have had strikes in the past - but are not involved in the current dispute over driver-only trains.
There is no direct tube to Waterloo from Kings Cross/St Pancras. You're on the wrong side of the Northern Line, so whichever way you choose, you'll need to change tube trains to get to Waterloo. So, what time and what day you arrive, and how much luggage you've got, will determine how fraught your journey from one to the other will be. If you're travelling at peak times (7.30-9.30am, 5-6.30pm) it'll be very busy and you might struggle to get yourselves, let alone luggage, on board at KX. Leave extra time for connections if travelling around these times on a weekday.
Under optimum conditions, allow at least 30-40min travelling time between getting off the train at St Pancras and boarding at Waterloo - that includes NO safety margins and doesn't allow for ticket purchases, delays, getting lost, that much-needed coffee/toilet/donut break. So best give it at least an hour ... Waterloo has a nice concourse with plenty of cafes, restaurants, shops etc if you need to kill time.
Be aware too that there's a bit of a walk from St Pancras to the Underground. I'd take other people's advice on where to change ... I think the Victoria and Northern Line platforms are close together at Warren Street, but I stand to be corrected on that.
Good luck. Oh ... and Weymouth station is on the eastern side of the town, so if you let us know roughly where you're heading, I can tell you how easy it'll be to get there.
although it's normally the front carriages that go on I've been on that train when it was the back coaches instead - threw everyone but there were frequent announcements.