From the passport section on www.gov.uk .........

"Time left on an existing passport is added to your new one - up to a maximum of 9 months."

So you can apply anytime within 9 months of the old passport expiry date and get the benefit of the remaining time off the old passport. Therefore the expiry date of the old passport is no longer relevant.

I will be renewing my passport this week, it has a September expiry date, I will be traveling to USA May/June this year.

Just bear in mind that it takes at least three weeks to get the new passport, April to September could be longer, unless you pay extra for "fast track"

https://www.gov.uk/renew-adult-passport is the official site, be aware of other sites that may seem to be the official site, but are not.