London to Liverpool as a day trip?

Is it physically possible? Yes, if you select a train that only takes 2 1/2 hrs to get there, and can schedule the tours between your arrival time at Liverpool Lime St and your departure time later in the day. Train stations are London Euston and Liverpool Lime Street. Factor in transit time to get to the station and to the tour start locations. Go to the National Rail website for specific train info and booking.

When are you going?

Right now the family home tours are at 10:00- 12:05 or 10:45- 12:45
This doesn't include travel time to/from Speke Hall or the Liverpool South parkway back to Albert Dock where the MMT leaves from.
Your last MMT tour would be at 2 pm from Albert Dock.

the average train time from London Euston to Liverpool Lime street station is 2hrs 30 minutes.
There is an early train from London at 6:16 am that arrives at 9:23 am. But then you would have to take a quick taxi to your pickup point of Speke Hall or Liverpool South Parkway, hoping to arrive before your 10:00 am or 10:45 am start.

Then of course 2 ½ hours back to London.

Only you can decide if that very long day is feasible. I wouldn't do it. At the very least, I'd stay overnight in Liverpool the day before.

Rather than backtracking from Lime Street to South Parkway (which the London train passes through non stop) change at Runcorn- arrive 0843, depart 0858, South Parkway arrive 0906.

My cousin is a Beatles fanatic. I'm not talking a big fan, I'm talking fanatic.

A few years ago, he and his family visited London. He thought he was so close to Liverpool he had to go. His family wasn't interested. So he decided to go by himself. He booked a Beatles tour and anything else he could fit in the day. Early train to Livepool.

Well, his train was delayed enroute and he missed his Beatles tour.

So, plan for the best but be prepared for the worst.

It’s possible, but luck and the transport gods would have to be on your side.
Why not travel up the evening before and stay over? As long as there isn’t a big event in the city a hotel shouldn’t be too expensive.