The nearest Airport is Leeds/Bradford. After a shuttle bus to Leeds railway station you will then travel to Beverley by train via Hull. 

Manchester Airport is approx. 2.5hrs away by train to/from Beverley. 

Beverley is on the rail network, on the line between Hull and Scarborough. London King's Cross to Beverley trains generally depart hourly via Hull and possibly via Doncaster. The journey takes 3-4 hours. Also from the North there are regular trains from York via Hull in 1.5-2 hours. 

See for timetables and planning. You are advised to book tickets and reserve a seat (and your bike) in advance. Tickets are usually cheaper when booked in advance.

Hull is only 10miles (16km) away and it takes approximately 25 minutes for a taxi to drive to/from the Ferry Terminal. Daily services run between Hull - Rotterdam (NL) and Hull - Zeebrugge (BE).