Welcome ScottTheNovice, lets start;
Unfortunately those children are going to be charged 100% of your miles as its going to be on a BMI partner flight. That actually leaves you in a bit of a bind;
4 x C+M Returns to the USA in business will cost you 150,000 miles and £840 in cash - you add any airline/government/airport taxes/charges onto that. To put that into an example - flying with Swiss from LHR via Zurich is going to cost (per person) £90 UK APD, ~£190 of taxes (fuel fees etc) plus a credit card fee of £4.50.
So thats a grand total of ~£2000 and 150,000 miles for 4 people (which you don't have, so you'd have to either buy more - you can just squeeze in the 24k a year and take you up to 150k, but that would cost you another ~£260 from BMI.
Better idea to get a few more miles: Get your partner to sign up for BMI and get a few free miles from whatever promo they have going and create a household account. Even better - you can also sign up the two children and do the same linkage to your BD account. Unfortunately at the moment there isn't a miles for sign-up promo (see this thread, there might be some alternative ways -
http://www.flyertalk.com/forum/bmi-d...ue-status.html). Even then you're still going to be short - unless you get a bmi credit card to get another 20k airmiles then you are going to either have to have 1 person take an economy flight for one of the legs across the Atlantic (and book as 2 1-ways, extra £4.50 fee) or buy miles.
I don't know if that figure blows your budget, as mentioned, if you go from Dublin (or anywhere else not in the UK) you'll save that £90 per person (total of £280), which if you travel out on a cheap ticket might just be worthwhile. Personally if with kids I wouldn't bother as the saving isn't that huge compared to 4 tickets at £40, a night in a hotel (so you don't miss your connecting flight) and more flying. The £90 will go up to £120 in November, but suspect that will be after you fly.
As for routes: Normally I'd say go via FRA/MUC/ZRH/VIE to get a much nicer flight from one of the European airlines - don't know if with young kids this would be attractive or not? If so then Swiss have their new business class on the ZRH-JFK route (very nice flat business seats on the A330 that would be a lovely treat - they are also great with availability). Unfortunately without a miles and more/expert flyer account checking availability is hard for Swiss.
The very best option from London would be something like Air New Zealand to Los Angeles, with great business class and from London. Unfortunately finding 2 seats is hard in business, I suspect 4 seats will be near impossible? Some of the US airlines will do trips straight from LHR and may have reasonable availability and will get you directly to the resort/city you are interested in most likely without needing a connection.
In case you are thinking of using economy tickets after those prices.. this thread will be very relevant/interesting;
http://www.flyertalk.com/forum/bmi-d...ee-ticket.html
To get the best value from your miles: Book as two one-ways and do 2 stopovers e.g. LHR-JFK-MCO and on the way back do MCO-PHL-LHR, so you can get to see Philadelphia and New York for the same price - you just pay an extra £4.50 booking fee. Again not sure if with your children this is a good or a bad thing to be going to different cities.
I can't really advise on US based resorts: There are too many really. I like the West Coast for heat, Disney is nice, San Francisco is also very pleasant: Its better value for your miles (longer flight that would be more expensive), however with young children the difference between 6 hours and 12 hours might be significant! I've written enough though, so I'll let someone else offer those suggestions