I try to avoid companies that are big enough to have an HR department. While I was discussing what I thought about a position at a company with their HR person I actually told her that the fact that they were big enough to have HR department was a strike against them. <g>
When I've gone to new jobs, they always wanted me to start as soon a possible so the OP's scenario wouldn't normally be a problem. It's never happened to me but if it did I'd just start 2 weeks sooner at the new job.
ProTip for meeting a coworker before a new job: maybe have an open mind about this person being helpful to you instead of spouting a prejudgement about their existence being a problem.
I meant no offense. Companies "that are big enough to have an HR department" are generally bigger than the size of companies I like working for. It wasn't a dig at HR. Replace "HR department" with "front desk receptionist/dedicated IT/security badges/have a cafeteria" if that will make my preferences less offensive.
When I've gone to new jobs, they always wanted me to start as soon a possible so the OP's scenario wouldn't normally be a problem. It's never happened to me but if it did I'd just start 2 weeks sooner at the new job.