My experience is that rain comes and goes in London and is usually not in the form of a heavy downpour, but one leaves one's hotel room without a rain jacket or rain coat and umbrella at one's peril. It is not fun to get soaked to the skin, especially not with the cool temperatures you may encounter on an overcast October day in London. The city's too large to head out unencumbered, with the plan to return to the hotel for rain gear later if it's needed.
Rather than basing my packing on averages, I look at 5 years' worth of actual, historical, day-by-day weather statistics on timeanddate.com:
London weather October 2022
Just looking at last year's data, you can see that it usually got up into the low to mid-60s (F), but not much before noon, and the temperature dropped rapidly after 6 PM.
Use the pull-down box at the right, above the graph, to change the month and year displayed. Use the Search box at the upper right to change the location.
I normally travel with just a rain jacket since a raincoat is bulky and wouldn't fit in my suitcase on warm, dry hotel-change days. On this year's Feb/Mar trip to Rome/Naples/Salerno with only 4 total hotel stays as opposed to my usual 20 to 30, I decided to take a lightweight, unlined raincoat. I figured it wouldn't be a major burden on that trip and I could shove it into my tote bag if I needed to when shifting hotels. At the last minute, I added my rain jacket to the suitcase because I didn't want to have to carry a full-length coat around with me, day after day, in case it rained.
In the end, on that particular trip, the combination was worthwhile. On days when rain was unlikely, I either wore the jacket (it was very cool before noon, and there was some rain on quite a few days) or carried it with me, tied around my waist. On days when significant rain seemed likely, I wore the raincoat--and was sometimes very glad to have it. It meant I didn't feel confined to my hotel room during really wet periods.
I wouldn't recommend that type of duplication to a carry-on traveler or to someone changing hotels frequently, but I'd do it again for an off-season, mostly-stay-put trip.