Collectively the family's savings are £60,000 so interest is only charged on a mortgage debt of £140,000 (£100k - £60k). However if one family member needs to withdraw a portion of their savings, this amount is then added to the original debt.
Conversely if any further money is paid into the offset account, the mortgage debt is further reduced.
The downside to this strategy is that nobody receives an actual income on their savings.
One final point - Only a small number of lenders will offer family offset mortgages. Plus, as the market is relatively small any deals might not be that competitive. Still, if you're interested they're worth investigating.