It's nice to see a farm that is comfortable enough to share its daily happenings online, and that people are supportive of it!
I haven't watched the clip (too many advertisements!), but sometimes cows go down with hypocalcaemia (calcium deficiency) after milking. You can remedy this, with bottles of calcium, if you're there quickly enough. There are also other problems, like vaginal prolapses, general calving complications and sometimes issues post C-section.
Most calves in the UK (providing that they're not suckler calves, i.e. beef calves left on their mothers) will be bucket-fed, once their mothers stop producing colostrum. (Note: calves whose dams die can be given defrosted frozen colostrum from another cow). There is a whole range of specially-formulated powdered milk for calves, so I'm sure that the calf in question will still grow up into a strong, healthy calf.