The best and quickest way is by train but to get cheap fares you must book approx 12 weeks in advance when the cheap tickets go on sale and commit yourself to a specific train. When will you be arriving?

As for the time to get through immigration - that depends on how many other long haul flights arrive at about the same time as yours. It could be as much as 2 hours though hopefully it will be less than half that. When booking advance train tickets, however, it is better to err on the side of caution and presume it will take a long time. With a 7.50 arrival it would be safest not to book a train from Euston to Lime Street before 12 noon.

I'm with chesterbuff, and wouldn't book a departure from Euston before noon. You may be through passport control and have your bags in about an hour, but your flight may be delayed in getting on stand such that you may not complete the arrivals process until 0930.

Travel time from LHR to Euston will be about an hour. There is a ramp down from T3 Arrivals to the subway [underground walkway] to the Tube station that serves T1,3; there are travelators along the way as well to speed the walk.

Board any Piccadilly line train towards central London, but sit near the rear of the train. Alight at Green Park station, and take the escalator up to the ticket hall - but don't leave the station.

Instead, take another down from the ticket hall to the Victoria line platforms for a train towards Walthamstow Central. Alight from that train at Euston.

The escalator adventure sounds unusual, but it is a much easier way to walk from one platform to another at Green Park - I do it frequently, with two rolling bags to pull. Following the directions in the station will mean stairs and long walks on the flat - not worth it.

Cash single fare is £5.30 for adults, £2.70 for children 11 - 15 and free of charge for children under age 11. If you plan to return to London and / or will use the Underground on a return journey from Euston to Heathrow, consider buying Oyster cards for each adult.

The cards have a refundable £5 deposit on each, but Oyster single fare from LHR to Euston is just £2.90 off peak (after 0930 on weekdays, any time at weekends or public holidays).

Not sure about the Heathrow to Euston part of your journey but from Euston railway station to Liverpool Lime Street you need to get a Virgin Trains service.

It's an hourly direct service and takes 2 hours 8 minutes.

There are other trains operated by London Midland trains that might be cheaper but they take a lot longer.

However, if you book well enough in advance on Virgin Trains you can get a single one-way ticket for as little as £12.00. These tickets are restricted to the train you are booked on though and are non-transferable.

As chesterbuff says above, aim to get a train after 12pm to be on the safe side.

Check out http://www.virgintrains.co.uk/ to buy tickets.

You can check times and fares on http://www.nationalrail.co.uk which will also show you the trains operated by London Midland; but Virgin Trains are quickest and the National Rail website will just redirect you to the Virgin Trains site anyway to buy your tickets.

Don't be tempted or talked into using any train booking website other than the two I have mentioned above.

Edit: Traveller Plus has answered your question on how to get from Heathrow to Euston. ;-)

You need to first buy a family railcard, visitors to UK are allowed to buy them - see website. http://www.familyandfriends-railcard.co.uk/

The railcard can be bought at the ticket office window on the day.

This gives 4 passengers with you a 33% discount on train fares throughout britain and your child travels for £5.

But more importantly there is a special concession with the railcard which will permit you to use off-peak tickets on Virgin Trains -the operator between London and Liverpool - even at peak times, so advance booking is just not necessary.

With the card your return off -peak ticket is £50 each from Euston - see if you can get a through ticket from Heathrow tube station (but only after you've bought the railcard).

Now you can just reach London Euston at your leisure and take any train, departs at xx.07 every hours with some extras between 5pm and 7pm.

Its just not worth the risk of missing an advance booked train, or waiting for hours because you allowed too much time for arrivals.

The advice posted above re change at Green Park is excellent. Also board your Victoria line (second) train at the rear to be near exit at Euston.

The Family railcard is a good idea - but you cannot get it at Heathrow (no rail ticket office) so it will have to wait until you get to Euston. No point in a through ticket as you won't be able to pick it up.

Add a flight to Manchester to your existing ticket then take the train direct from Manchester Airport

Traveling into London and taking a train to Liverpool will be quicker than checking in again and flying to Manchester. Manchester Liverpool train is not a good experience in my opinion; journey is still 1hr 15 min at the best whereas London train is under 2hr 15 min, but admittedly you will save some trainfare (or could take a taxi from Manchester).

Which is why I said add a flight to their EXISTING ticket which means not having to collect luggage, nor having to check in again. It also means they could have a short connection time and be in Manchester whilst still in their way to Euston and if things go wrong then it is the problem of the airline to sort out rather than having to buy a new ticket at last minute prices

"Manchester Liverpool train is not a good experience in my opinion"

I agree with that statement - it can be hell, especially if Northern Rail are involved !

Actually, add First Trans Pennine and East Midlands trains to that too, especially when the trains are busy - which is most of the time !