"Nickel and diming" is usually applied to someone or something (like a big company) that has a significant economic advantage over the person or entity it is doing business with and that exploits its advantage by adding a series of niggling fees and penalties to the ongoing contract at the expense of the economically weaker party. A typical example is a bank that adds various processing fees to its interactions with account holders—a teller fee for transactions that an account holder conducts in person at the bank; an ATM fee for transactions that the account holder conducts via ATM; and so on. The terms that yield all those nickels and dimes aren't subject to practical negotiation between the parties; the financially more powerful entity simply imposes them by fiat, and the weaker party either submits to them or looks for work (or a business relationship) elsewhere.
But the situation you're talking about is the reverse case: the employee or contractor or weaker economic party builds up the volume of work that he or she charges for by painstakingly including any fraction of time spent on any task arguably related to his or her employment. The ultimate chutzpah in this situation might involve treating the time that the worker spends recording each tiny fragment of billable work as billable work.
In one sense, this process does amount to nickel and diming, though with the significant difference that the more economically powerful party can usually cut its losses fairly easily by finding someone else to do the work. But given the somewhat unusual stronger-to-weaker flow of the nickels and dimes, I would use a different idiomatic phrase to describe what the billing party is up to: either "always on the clock" (which refers to the punch-clock that records reimbursable time at a job site—or figuratively, billable hours in, say, a law firm) or "always has his [or her] meter running" (which refers to the running tab that a taxi driver charges for transporting a passenger somewhere, whether the cab is moving or not at any particular moment).
The suggested phrase "padding [one's] time sheets" seems to me to describe something fundamentally different—not "charging for every minute of service that one performs on a client's or employer's behalf, regardless of how negligible a particular itemized action may have been," but "exaggerating the amount of time the person actually spent working." The former is perhaps (to the client/employer) a form of aggravating and expensive punctiliousness; the latter is a form of theft.