As others have already indicated, all bank accounts are entirely separate (unless you've actually signed up to an arrangement linking two accounts together).
I'm currently paying back a debt to Abbey (relating to a closed account) through a debt collection agency. At the same time I've got money in a separate (active) Abbey account. Abbey can't take money from that account to pay off the debt because they have no right to assume that the money held in my account is actually owned by me. (I might, for example, have opened an account in my name to hold money held for other people).
However, your wife's credit rating will almost certainly be affected by your debt. Credit reference agencies are entitled to assume a financial 'association' between husband and wife (and between other family members resident at the same address). If your wife applies for credit she may well find that it is refused because of your excessive overdraft.
Chris