Poor people suffer worse health and so in the UK they will be visiting the GP.
In the USA poor people also suffer worse health but can't afford to visit a GP so they don't.
= shorter waiting times.
You don't normally visit a GP in the US unless you have HMO insurance or need a referral for a specialist because you don't know if you need, for instance, a hip replacement or physiotherapy. If you have Medicaid you will also be in a HMO-type situation.
Those with a PPO can go straight to whatever specialist they think they need. Psychiatrist, OB/GYN, ENT, general surgeon, neurologist, oncologist, etc.
America has a shortage of doctors in general. Some rural areas are particularly badly hit by the shortage. Cities have little independent storefront clinics and also for-profit chains, treating mainly Medicare patients, advertising heavily for clients, but appointments are available fairly easily.
The very poor in the US have Medicaid or Medicare, and there is also the VA system, which is for veterans. It's the people who are a tier above the very poor who are really badly hit by the 'insurance industry model masquerading as a healthcare system' model. People who have jobs that offer crappy health insurance (60/40) or who have jobs with hours and conditions cunningly designed to let employers off the hook for insurance provision are really squeezed.
Your wait time will depend on the specialist you want to see, in general, and also the location of your doctor's practice.
For instance, my family's pediatrician office was headed by the professor of pediatrics in a local urban university hospital, and the waiting room was always heaving.
For one frustrating year we took our business elsewhere, closer to our suburban home, to the office of a pediatrician who had admitting privileges in a nearby for profit hospital. No waiting times. No decent phone triage either, so they would blithely tell you to come in and have the doctor take a look at whatever your child was complaining about, tell you it was probably a virus, and bill you.
By contrast I had to almost beg and plead with nurses over the phone to get to see a pediatrician in the university clinic. While this may sound horrible, it actually saved me $$$. And over the years, with little or no nurse turnover, the staff got to know me and trusted my assessment of whether my child had an ear infection, etc.
I had a local GP until two years ago. I had strep and couldn't get an appointment until ten days after I called to croak over the phone that I needed to see a doctor. I walked in at the local Walgreens in-store clinic and got the throat swab there from the nurse practitioner.