Weather

You need to find a site that lists historical temperatures by city and figure out the averages. October is a month when things start really cooling down and it will generally be warmer Oct. 1 than on Oct. 31. Also look at a map and London, Paris, Budapest are much further north than Italy or Greece. The more northern cities will definitely be cooler, but your comfort level will vary based on what you are used to. I am from Houston, Texas so I would want warm layers for some of those places. Italy and Greece will be warmer but may still have cooler evenings or rainy days that are not as warm. Yes layers will be important.

As someone just posted today in a different thread, at some point in October the weather will probably change. The problem is that we don't know when that will happen. You can look at historical day-by-day temps on wunderground.com now, then when you get inside the 10-day mark, you'll have access to predictions that may be more accurate than historical averages. (Or not.)

I think you'll avoid true cold weather, but it can be cool and damp in London even in mid-summer, so I'd definitely want layers in London. I have been uncomfortably chilly in London in June, though that was before global warming kicked in. Paris and Budapest are also risky, in my view. But your planned sweater and coat should be more than adequate as warm layers. It's unlikely that you'll need a heavy coat.

I think multiple dresses plus two pairs of shorts are probably more warm-weather clothes than you'll need, but they don't weigh much, so not much harm in them. I would want one or two long-sleeve tops for sure.

I'm more concerned about the shoes, though. You'll likely encounter rain; will those shoes be OK on wet cobblestones and marble steps?

My favorite time to visit Budapest is from about the third week in September until the third week in October. You will find us there almost every year at this time. It’s a little cool, but the outdoor cafes are still open and they have heaters. The weather is pretty ideal for walking and sightseeing. Dryer than much of the year as well. But that’s all antidotal. If you really want to know the weather odds you need to visit a site like this: https://weatherspark.com/averages/32212/10/Budapest-Pest-Hungary

As for what to wear? Mostly what you would wear in your own home town for similar activities during similar weather……. To travel lighter and have a little more flexibility I use thermal underwear and a layer or two in lieu of a heavy jacket.

My first visit to Florence was in mid-October and I was so glad I had booked a room with good AC. I once went to Paris in mid-September with only sandals and my freezing toes made me buy sneakers and socks on day 3. The weather is unpredictable. Be prepared, though I doubt you'll need more than a fleece jacket, you're likely to want both sandals and closed shoes and rain gear.

Historical weather patterns won't help you much, since Mother Nature seems to be angry with us. Especially in Great Britain and France, weather systems change at the drop of a hat - the morning predictions will usually change radically by the afternoon.

The best advice, since you will be travelling for a month in so many different areas, is to bring thin layers and a waterproof jacket. It's highly doubtful that really cold weather will begin until November.

I use the Norwegian met service yr.no for climate data. However, while it reports temperature and days of precip/month data by month, it is limited to mean (average) normal, high and low temperatures. October data example for YR.NO London for October:
12.4°C normal mean;

15.3°C mean high;

9.6°C mean low;

9 mean days of precipitation in October
Where a rain day means more than 1 mm but doesn't differentiate between 2 mm and 2 cm of precip.

Some countries like the UK have more detailed climate data reports that in addition to means, report the extremes. The extreme data gives you a better sense of worst case weather. October data example for the UK MET OFFICE Southern England:

This describes the main features of the climate of Southern England
from Kent westwards to Wiltshire and Dorset Much of the eastern half
of this area is densely populated, as it includes Greater London....
Reading the temperature graph for Heathrow in October:
24C Extreme High
15C Mean High
8C Mean Low
-5C Extreme Low

What this data set shows is October in London or Southern England is fall-like weather. You could get days of frost followed by what we here in the States call a First American Summer.

Could be warm. Could be cool. Could be wet and windy, or not. And keep in mind the old saying that a bad day of holiday travel is better than a good day working.