A quick data point here - I renewed my passport last/this week - from applying on line to receiving my new passport was just 7 days,

I would recommend using a photo-me booth to ensure your photo meets the passport office photo requirements and sending in the old passport using royal mail guaranteed next day delivery.

My offspring in the USA renewed their passport at the end of May online and it took five weeks from start to finish (delivery), but a good bit of that time was documents in transit.