I warn you I score about 1/10 in the 'knowledge' stakes, but I guess you'd like some kind of response before bedtime at least! [My very best suggestion is at the bottom of this page!

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If it's 4-5cm it's not a baby, I'd have thought. How long's the fur?
Anyway.
Make absolutely sure some of that milk gets down its throat, no matter what process you use. Don't dilute it too much, maybe add a grain or two of sugar. Make sure it's warmer than fridge-temperature. I don't imagine you've got any goat's milk or colostrum in your larder?

Joking, but most mammals prefer them to cow's milk.
Shavings of chocolate, rice crispies, oats (i.e. porridge-type) or any other cereal would be gratefully received when it's fitter and hungry. Also a chopped, unsalted peanut.
Ensure it's in a fairly solid box, or you'll wake up tomorrow and find a hole in the cardboard, and an unhealthy mouse somewhere in your house about to die under the floorboards, with the subsequent putrid smell etc. If you're to fix the mouse, make sure it can't escape just yet.
Put a small sock or bobble hat in there which can get filthy, as will the rest. Also a small cardboard box to act as a 'kennel'. Make sure it's somewhere dark, quiet, undisturbed by pets/kids, and not too warm - the airing cupboard would be wrong!
If it's wobbling it could be dehydration, so water and milk's correct.
Best leave it alone with all the facilities overnight or so. Don't intervene too much or it may be too panicked to eat, drink and rectify itself.
Good luck, and well done for caring! If it doesn't survive,
do not take it personally!
[Hope that you'll soon get some decent advice from someone else who's less of a rookie than me!!]
EDIT: Don't put it somewhere COLD though, as it could be a bit of hypothermia. The cotton wool etc will help keep it warm, and make sure the newspaper's torn and bundled up for insulation.