1. In terms of costs, return airfare for an adult to London is typically S$2K+ and child ticket is about 70%-80% of adult fare. Airfare alone will probably set you back at about S$9K for the family. In terms of accomodation, food and sightseeing, it really depends on how much you are willing to spend. Min. food cost per person is about £5 (S$10) for a meal. You can use that as a rough gauge.
2. There is a lot to see in London and depends what kind of holiday you want.
I would recommend these free places for a starter:
a) British muesum
b) Buckingham palace
c) Borough market and London bridge
d) Harrods shopping
e) Oxford street shopping
f) St Paul's Catheral
g) Greenwich GMT meridian Line
h) Westminister Abbey and 10 Downing street
i) Picadilly circus and chinatown
In terms of accomodation, budget hotel includes travelodge with family rooms but I think you probably need 2 rooms. Alternatively find someone in London to stay with, accomodation in London can be very expensive.
3. Winter clothings - suggest you buy them in London, cheaper and nicer.
Hope above helps and enjoy your trip to London. If you need any more advice, free feel to msg me. Thanks.