By "a gap between the worktop and appliances" do you mean between the top of the appliance and the underside of the worktop? The appliances have adjustable feet beneath them - are you sure the fitter hasn't jacked the feet up? One thing you will need to do is to check the height at the top of the units. In a standard kitchen this should be somewhere around 870mm (150mm of plinth + 720mm of base cabinet). What height are your cabinets now?
That 870mm normally gives you 20 to 40mm working space between the top of the appliances and the underside of the worktop (with my Bosch appliances about 30mm - our washer and dryer are both about 840mm high). Are your kitchen units lower than that? If no, then lower the feet on the appliances (they are threaded). If yes, and the appliance feet aren't jacked up, then the cabinets will need to be jacked up a bit to give you the required clearance; the cabinet feet are adjustable, but the cabinets may also have been fixed to the wall (screws, angle brackets, etc) so they will need to be unfixed, lifted, levelled then refixed, the plumbing for the sink will also need to be altered (see below) and you'll need to replace the decor end panels as they will end up cut too short (sorry, but joinery and kitchen fitting are subtractive trades - you can't weld extensions on!).
As to to sink, there are two main connections - the taps and the waste. The taps hopefully won't be an issue, because most taps are fitted on flexible hoses these days, which gives you a bit of leeway to move the sink. The waste will in all liklihood be 40mm plastic which may need a straight section cutting out and replacing with a longer one - waterpump pliers, hacksaw and deburrer job if the waste is push fit or compression fit (for a DIY job I'd do the same if the original were solvent weld pipe, the only difference is that it would need a couple of extra straight connectors). Your biggest problem is likely to be freeing the sink from the cabinet - Belfast sinks are normally bedded onto the cabinet with a generous amount of silicone sealant and that will need to be cut, which can be a PIA job. BTW the infill panel to the right of the sink with a bare chipboard top is asking for trouble - all cut edges around the sink need to be completely sealed with something waterproof (e.g. silicone, etc) before the top is installed (there is no guarantee that the granite fitters will do the job - I assume the tops are to be granite from what you wrote plus the fact that it's a Belfast sink)
Basically, if the appliances can't be lowered, you are looking at a partial refitting of the base cabinets plus new decor panels. I think your fitter has stuffed up the job and should be redoing thIngs at his own expense - I doubt he'll like that, though