The best way, in my opinion, is to look to see which card has the highest interest rate and then pay minimum plus £420 per month for that card until the debt is repaid, then pay minimum plus £420 per month on the other card. You should get both the debts paid off fully in about 7 months.
There is no point in using savings as you've suggested unless the interest that you earn on your savings is noticeably more than the interest that you are paying on your credit cards. This may be the case if some or all of your debt is subject to a "0%" or similar promotional deal.
The transaction charge would mean that starting with a new "0%" promotion on another card might not be worthwhile.