Been through similar and more than once over the years.
I had spares clothes here, new underwear, socks and toothbrushes bought cheaply. And always stuff like deodorants around.
I put in an everyone has to shower before bed rule, including all adults. One of the teens would then grab all their clothes and chuck them in the washer. Whilst having a chat with a friend so they would follow them into the kitchen. And at some point, they were told they could use it whenever they wanted. Some were embarrassed at the thought and said something like oh it's fine mum won't even realise who's clothes are in the machine.
My dc's would have the chat as they knew their mate better and how they would respond.
The basics were - make no promises only that you will do what you can, can talk to my dc or sibling (close in age) or myself about anything, ask how things were at home. And that there were times when an adult would have to be brought into it.
My dc's were always upfront when a friend told them something and they weren't sure, that they would talk to me, or the mate could come directly. The friend was always reassured that if they wanted it could be an anonymous chat to start with. And they would come to me for anonymous chats.
It was always handled sensitivity without drawing additional attention to the teen. They know they smell. They get told every single day. In time they come directly to you as trust is built especially when there's any type of abuse at home. They'd talk about health because let's face it if clothes aren't being washed what's the likelihood of them seeing gp etc. And being overweight there might be health issues going on. Mine basically did their usual daily things and drag their mate along, instead of let's meet at 5 after my gp's appointment, they'd come home first, get showered and changed and of course, the mate didn't want to be in the uniform. Which gets the uniform in the machine as I didn't always have school night sleepovers.
I also made sure that my dc's were also supported when dealing with these things. Remember until that friend starts to trust no-one really knows what's going on, and hopefully by which time you are already involved directly. Not that I am saying theres really bad things going on, just mentally prepare yourself for just in case. Have some type of plan in place for just in case, so IF needed you appear to be in control whilst on the inside you aint,
Each teen is different of course, and there are moments when you ask yourself wtf are you doing.
Talk to the school of course. They will already be aware. However, even if they did refer to the family link people if mum says no then nothing can go ahead. Depending on additional info, a referral to SS will be made, you can also go down this route. Don't feel guilty if you have to make the call, there's only so much you can do. It's nice that you care enough to help rather than go down that route. It could be a temporary thing at home, and they just need a bit of support. It could be more and by opening dialogue you can help them contact SS. Going the SS route is hard for teens, and of course all children. They are dealing with all the usual teen stuff, with stuff going on at home, and these strangers wanting to know everything about them, and few if any people they can talk to.