This is just a personal opinion, but I suggest 'slow down' is more appropriate where one has the option of NOT slowing, e.g. in a car, while 'slow up' is more for where one does not have the option, as in ageing.
Phrases like this often change over time. In the past, when something was someone's responsibility, we used to say: "It's up to you." Nowadays, it's probably more common to hear: "It's down to you." As in the example you give, these would seem to be opposites, but they are - effectively - synonymous. Neither, now, is any more 'correct' than the other in general use.
I'm with Hippy on this one. 'Slow up' does appear to more the American usage. Same as Americans prefer to 'fill out' a form where we would say 'fill in'. But I think it's down to personal preference.
Just an observation, but since both mean to decellarate, it would seem more logical to use the British 'slow down', as your speed does indeed go down, not up.