What is the cheapest way (by air) from HK to Newcaslte in UK

You can try:

http://www.opodo.co.uk/ or http://www.ebookers.com or http://www.ba.com to check the price first, then contact travel agent to get a similar quote:

For example, via KLM, HK --> Amsterdam --> Newcastle is GBP 666.90 including tax on www.opodo.co.uk

Outbound: Mon. 25 June 07 Inbound: Mon. 02 July 07 Highlight
Departing: 10:20, Hong Kong International, Hongkong Departing: 06:05, Newcastle International, United Kingdom GBP 666.90 **
Arriving: 17:55 , Newcastle International, United Kingdom Arriving: 08:30 + 1 day(s), Hong Kong International, Hongkong GBP 666.90**
Total price including tax and booking fee
Duration: 14:35 + 1 stop(s) Duration: 19:25 + 1 stop(s)
Carrier: KLM Royal Dutch Airlines Carrier: KLM Royal Dutch Airlines

Alternatively, If you want to sight-see in London as well, you can travel to London Heathrow first, then buy cheap ticket via http://www.virgintrains.co.uk/default.aspx

Another way is to travel on coach from London Victoria.

http://www.megabus.com/uk/
or
www.nationalexpress.com/