I live in a cold climate where it snows and can be very cold so I understand. We still always rug up and get out there for a bit but there are limits. We are not one of those towns that has play centres like flip out or Monkey Mania so home it is or with friends.
Building forts with sheets and pillows is a good time filler. a sheet over the dining table is so easy.
There are some great you tube videos like “Danny Go” where the kids have to copy the dance actions. It gets them moving, improves motor skills, and is great fun.
Having craft and play doh etc easily accessible in tubs is a must.
Go to the library for reading day, playgroup etc. Some towns are lucky enough to have little playgrounds in shopping centres where you can have a coffee and watch them play.

As a former childcare worker ..you can do all the same things you do in summer as in winter. Paint on the patio using an easel or table. Set up an obstacle course on your patio. Gluing, colouring, playdoh, cloud dough, goop and slime all outside on patio and they can enjoy the rain whilst doing an activity. We once put raincoats on the kids and either gumboots or plastic/rubber gloves on their feet and told them they were ducks on a pond. Much laughter and fun was had and they all wanted to take off their boots and wear the gloves instead. Let them help to make morning and afternoon tea with you. There are plenty of free places to go and libraries sometimes have special activities on for all ages. Walks in the rain to the park or around the block in rain gear, jumping in puddles and playing with mud pies is always fun and good for their tactile development. The things you can do are only limited by your imagination!

Is he into painting? Can you keep all your left over yoghurt tops, toilet rolls etc and do some craft and painting with them. You could also get potatoes cut them in half and create some art work. Other idea is to make paint out of yoghurt and food colouring and put your toddler into the bath with a paint brush and paint and let him go wild painting the bath. My daughter loved doing it at this age.

I always used to have indoor Picnics the rug plastic wear food drink and toys and they loved it

Library rhyme time, trampoline places have toddler areas too cheap days for under 5s that have a separate area with ball park etc, just call to check. Same as alot of indoor play centres call to check. Look up indoor play centres.

My kids used to love going to the museum- there are usually lots of interactive things for kids there- my local museum has an inside and outside area for kids to explore. They also loved indoor playgrounds- while that’s not outside, a lot of the equipment is things they might enjoy at a park, so I think your little one will soon find a new love!

Coming from the Netherlands myself and having lived for many years in Northern Ireland, cold and wet weather never held us back to go outside. Rain coats, rain boots, warm coats, scarfs hats and gloves was certainly a must at times. We visited parks and made walks and made cycle rides all year around. But of course there’s time for inside play as well. Obstacle courses, craft time, jigsaws, reading, making music and dancing, we visited indoor play centers as well

We do a combination of outdoor and indoor time during the colder months. We layer up for the colder months when outside. We do plenty of craft and creative inside activities when it gets really chilly! Cooking is terrific during the cooler months.

I would suggest a play centre or an indoor cafe with a play area for kids.

Layer up and go for walks when its not raining. And when it is, go to an indoor play area or do stuff indoors that he likes maybe adding nature elements if that makes him feel better? Like leaves bracelet or nature loose parts and playdough or nature collage on clear contact paper

We are doing all the same things as summer just with layers on. Walks to the park, chasing the seagulls, swimming lessons (heated indoor pool), gym classes, bubbles in the carport, chalk play on the driveway, gardening on the back deck. We seem to do more pillow forts now that he’s climbing on everything.

Does your little one like play centres? There are a few good ones around where you can watch them so their own thing while you have something to eat.
My kids are 2 and 3 and love Timezone lol but your LO might be a bit too young at the moment.