Is there any direct train from LHR Airport to Newcastle Gateshead?

Take the Piccadilly line to King's Cross St Pancras, and change to an Edinburgh-bound train taking you to Newcastle (there are also Newcastle-bound trains, but those are much slower). Then catch the metro to Gateshead.

An alternative route which might be better if you have lots of luggage and small children would be to get the Railair bus from Heathrow to Reading and then get a train to Newcastle. There are direct trains and also some which involve a change on route.

A very quick search with Google suggests that a flight might be viable.

British Airways operate several flights each day from Heathrow to Newcastle with fares as little as £92 (Google list), while LNER run trains from King's Cross, starting at £160 - you might get a lower price with advance booking.

A direct flight takes just over an hour and will save you time travelling into London, plus three hours on the train.

Heathrow does not have a mainline rail station so there is no possibility of a direct train to Newcastle.

The most common route taken is the Heathrow Express to Paddington, switch to Kings Cross by underground (or taxi if you have luggage) and take the direct train to Newcastle. There are trains about every 30 minutes most of the day.

With four of you a taxi from Heathrow to Kings Cross is also viable, and a good bet if you have luggage. The Piccadilly line is cheapest but slowest, and very hard with luggage.

Just for completeness I will mention the existance of National Express Coaches, probably the cheapest mode of transport from Heathrow to Newcastle. A ticket would cost £30.00 per person.

The downsides are there are no direct coaches from Heathrow, you would need to travel to London Victoria Coach Station and change there, and the time it takes to travel around 7hrs.

Fly. It's much faster and likely cheaper.

The LNER train from Kings Cross to Newcastle is only cheaper than flying if you book seats on a specific train, which you realistically can't do if you're flying from India - if your flight is delayed or you're held up in immigration then you'll lose your tickets.

The flight takes an hour, the train will take closer to five once you factor in travelling time between Heathrow and Kings Cross.

If for whatever reason you have to take the train there's also the problem of getting from Heathrow to Kings Cross at 7:30am on a weekday - either brave a long rush hour tube journey with your family and all your luggage (never nice after a long flight), or get an expensive taxi.

For rail enquiries in the UK use the National Rail site. As other answers have pointed out, trains to Newcastle leaves from King's Cross station. The cheapest fares are if you buy in advance for a specific departure. However, that can be risky when coming in from a long flight with risks of delay, as the tickets would be worthless of you miss the train.

With 4 people, a rental car could be the cheaper option, but perhaps not such a good idea after a long flight and jet lag.

Best would be if you could get a connecting flight added to your ticket. Might not be possible at this point.