Put the sling baby on the back if you are struggling with a toddler. Breastfeeding the baby, you choose - either take the baby off and deal with the toddler, continue feeding the baby and deal with the toddler (sling can help if you use if for breastfeeding) or continue feeding the baby and help the toddler afterward.
Don’t put her in layers and layers or waterproof clothing. Nice thick leggings or joggers. Leg warmers under if necessary. Have spares with you. Including socks. Wellies will wipe. Leg warmers and dresses or long t shirts for at home.
Remember many toddlers are unconcerned by accidents, sometimes that’s part of the problem. But really sometimes eg at the park they can be pottering around for ages and it’s not till you pick them up, spot a telltale patch etc that you realise they’ve had an accident. Once you realise it’s possibly less urgent a problem than you might be imagining. Remove bottom half clothing, a wipe and replace with fresh. Bath every night.
Training pants also available, will contain small accidents but look like pants. Pad for the car seat.
Take to toilet at predictable times. Eg always after eating, always last thing before leaving the house, straight away on arriving home, just before bed.
Buses, they just have to wait or you get off and find a bush, no way round that one, if you are especially worried you can take the pad from the car seat for her to sit on (same if relatives houses/sitting on sofas etc if you are worried). Reduce risk by going immediately before and after bus. Centre of town, much easier than random suburb as there are toilets in towns. Download the find a toilet app and always have a 20p piece with you. Many places are more sympathetic to a toddler than an adult, rightly or wrongly. I’ve been refused many times for my toddlers but I’ve also been accommodated.
Learn how to hold her up for a wild wee. There is a technique. Turn her away from you and reach around to lift her knees in a seated position, then angle her away from you. Keeps everyone clean and dry.