The rules will be the same as for any other kind of pleasure craft, such as yachts. I would suspect that with a craft that small, it would need to be accompanied by a support boat, and so the rider would be treated as part of the crew of the support boat (I believe this is how it is done with cross-channel swimmers etc – obviously they cannot carry their passport while swimming!)
You would have to notify the relevant authorities (on both sides) well in advance, as the English Channel is the busiest stretch of water in the world, and small craft are at significant risk, as outlined here:
You must meet the legal requirements (COLREGS) of any channel crossing to make sure you’re not putting yourself or other people at risk. The strait has strong tides, sandbanks and shoals. Weather conditions can change quickly. Visibility is often poor and it can be difficult to navigate.
Large vessels passing through the strait can be difficult to manoeuvre. They can take several miles to stop or turn. Some container vessels are 318 metres long, 42 metres wide and service speed is 24 knots (35 mph). They have to commit to a course of action long before they can see a swimmer or detect small craft on radar.
This RYA Page suggests that you do not need to notify UK customs if you are entering from another EU country:
If you are departing the UK and going directly to another EU country, you are free of formalities with UK customs.
If you are arriving into the UK directly from another EU country there is no need to fly your Q flag or notify customs of your arrival unless you have goods to declare or non-EU nationals onboard.
This Government Customs Notice says that:
Anyone on board who is not an EU national must get a UKBA officer’s permission to enter the UK from a place other than the Isle of Man, or the Channel Islands. As the person responsible, you must make sure that anyone requiring immigration clearance (including yourself if appropriate) obtains the necessary permission to enter.
If there is anyone on board requiring immigration clearance, they will need to contact the nearest UKBA office by phone to arrange clearance. The National Yachtline will be able to advise on how to do this.
Noonsite, however, states that
As a member of the European Union, EU regulations apply. In principle therefore boats coming from another EU country do not have to clear Customs, although they have to report to Immigration.
It also suggests that the requirements have tightened up recently, and I’d imagine they will tighten up further in the next few months, as there are currently many reports in the media of people-smugglers trying to use small boats to bring migrants across the channel illegally. It may well also change after the referendum this month…
It probably also depends on whether they actually enter France, or merely load the JetSki straight onto the support boat and turn round to return to the UK.