As other have said, DFDS Newcastle - Amsterdam may be an option, depending where in Yorkshire you live. According to their website, they are taking bookings. I have used that crossing, but, not with a bicycle, so, cannot advise ease or otherwise of booking.
The service runs daily, leaving Newcastle (North Shields to be precise) at about 17:00 and arriving Amsterdam about 09:00. When booking a cabin, I would advise paying a little extra and get one above the waterline, in our experience, the cheapest below decks cabins are not compatible with a nights sleep (poor ventilation, and, too much engine noise).
Two final points,
if you are a Tesco shopper, you can get discounts on DFDS Services, using Tesco Vouchers
it is worth a phone call to their booking office to check for deals, on one occasion (with a car, and heading for Sweden), we bagged a return crossing deal, with camping vouchers, for less than the scheduled one way fare!