Get a puddle suit for the toddler (should be able to pick one up on Amazon Prime) and some sexy waterproof trousers and wellies for you. Raincover on the pram and go to the park/for walks/to look at shops. Pets at home is good just to have a mooch round, talk about what you would choose for your fictional pets, see what noise the different rattling/squeaking dog toys make, admire the animals (a good lesson in being respectful and calm around them, or you leave) feel the textures of the different pet beds (make sure hands are sparkling clean first!)
Do you have a library within walking distance? That's always a great wet weather option
For at home, a sensory-tray-a-day helps keep the boredom at bay - dried rice (you can dye it if you have food colouring and hand gel) pasta, jelly, sand (if you have a sandpit you can just bring some in and have it in a shallow box in the kitchen instead. If the sandpit sand is damp dry the indoor sand out, or vice versa), beans (NOT kidney beans, they're toxic raw) etc etc.
How old is the baby? If big enough to sit up and join in they can make a horrible mess in an empty bath with 'paint' made with food colouring and Greek yoghurt with a couple of paintbrushes. Strip them down to nappies and have them sit on an old towel so they don't slip over and let them go nuts. You can rinse out the bath and wash them off afterwards without them spreading mess everywhere through the house first.
For the future, invest in a little folding slide that you can bring out on days like these, preferably with a ball pit (if you have an inflatable or framed paddling pool that would probably work) for them to slide into.
Have a living room disco! Draw the curtains, drape flashing fairy lights around and pump up the music. You can hold the baby and have a good old dance too.
Have any old birthday balloons in a drawer somewhere? Blow one up and make a makeshift racket with a paper plate and a wooden spoon and show your toddler how to play balloon tennis.
Make a den, but somewhere different. If you always use the drying rack when your toddler wants a house, maybe make a different one by spreading blankets over the kitchen table instead.
Do some baking together.
Again for the future, but if you're ever going to throw away an old mattress consider keeping it (under the bed perhaps, or in your garage if it's dry and rodent free!) and bring that out for an indoor bounce station in desperate times. Surround it with pillows/duvets etc so they don't hurt themselves if they fall off. When they're a couple of years older, if you have several single mattresses you don't mind getting a bit ruined, you can line them up the stairs with another mattress/lots of sofa cushions and other coverings at the bottom and let them play mountaineers climbing up and falling down - they can literally spend half a day doing that while you put your feet up.
If you have any wrapping paper tubes you can make a giant 'marble run' with a golf ball or similar. Even better is guttering pipes and a larger ball or small cars. Honestly set that up down the stairs or at a slant somehow and they'll play for ever so long.
Have an extra long daytime bath. Add random household stuff they wouldn't usually use and let them just explore it for hours. As they get older a blackout blind for the bathroom window, a UV lamp and some fluorescent facepaint goes down well!
Let your toddler make a collage with all the random bits of different materials you can think of/find. Old scrappy bits of wrapping paper, kitchen foil, snippets off odd or worn out socks you were going to throw away, popcorn (you can pop your own quickly in the microwave, but don't let your munchkin eat it, it's a real choking risk), string, an old lolly stick, any sort of old gubbins that would stick to card with glue.
Move the furniture round in the living room. This is infinitely exciting for toddlers and the baby will enjoy it too if at the crawling stage. You can take the opportunity to vacuum under sofas and things, and maybe make a simple obstacle course too.
How about a really simple treasure hunt? Maybe 5 or 6 large shapes (very large for a toddler, using most of a piece of A4 card) that you 'hide' in plain sight in other rooms, and then go round the house together to help find them.
For future reference again, don't have access to all of the toys all of the time. Limit them to about a third of their whole collection, and rotate them to keep thing fresh and interesting. They're more likely to play longer if you're stuck indoors in abysmal weather if they haven't seen those toys for a while.
Have baby in a sling and let the toddler 'help' with chores! They can join you sorting dirty washing into whites and colours, and when it's been washed peg it out on a line you've strung at their height across the kitchen (light stuff like socks and knickers anyway!) if you give them some pegs. Let them have a duster of their own while you polish (some mums give a spray bottle of water too, but I'm not quite brave enough for that!) 'Helping' wield the vacuum makes it take 45 minutes longer but at least they're not stuck in front of the TV!
Have another mum over with her tot.
But yeah, get out of doors when it's not really hammering down. You'd go mad else!