Book in advance for Bletchley Park?

I don't know for sure, Lola, my experience was with a pair of free tickets on a regular September midweek day so May might be different, but probably not a lot different.

We arrived by car about an hour after opening and were easily able to park near the entrance. No queues - maybe a minute or two - to speak of to flash our bits of paper, and in we went. The school groups arrived before lunch and they did descend on certain buildings as they had projects to complete and got a teeny weeny bit loudish at times, but generally we went where we wanted when we wanted with only reasonable crowding at various pinch points.

If I were in your shoes I'd risk it.

Thank you Nigel.

And I see "day return tickets" at £20.20 for Off-Peak travel. Is there any reason to purchase these in advance? Or will the walk-up price at Euston Station be the same?

Off Peak Day Returns same price at walk up, but the little window down at platforms 8-11 gets long queues. You could buy using your phone if you have an account on any British train company or want to use the London Northwestern Railway app and get them there. You could also get them at the station at the Virgin ticket hall just to the left of the left hand entrance to the station or if coming up from the Tube, behind you as the escalator reaches the top.

Or get them as you pass other stations in your journeys around London or the UK. Just be sure the put the right date on the ticket. I've more than once sold a ticket with the wrong date.

Off Peak Day Return is valid leaving London after 9:15 and return that day. If they show ‘West Midlands Trains only’ they are much cheaper than the alternative ‘any permitted’ and allow you to travel on West Midlands Railway or London Northwestern Railway trains but do not allow travel on Virgin Trains. To Bletchley the best train is the xx:13 which only stops at Leighton Buzzard before Bletchley. The xx:24 and xx:54 are much slower.

I visited late August on a Friday, arrived via train around 10:30 a.m. and there were perhaps 2 people in line ahead of me to get in--I had pre-purchased my ticket and they do scan it, but in my case there were no lines to avoid.

The £20.20 Off Peak Return is probably the best ticket unless you and your travel buddy have Two Together or Senior Railcards which could lower them an additional 34%.

There is a Super Off Peak return which is £5 less (also eligible for Two Together or Senior Railcard) but it has more restrictions including leaving after 10:30 from London and on the return not arriving in Euston before 13:00 and also not travelling between 16:30 and 19:30.

When we visited, it was in a September and there were no lines at all.

I went to Bletchley Park before “The Imitation Game” seemed to elevate its popularity and even so, I can’t imagine there being a long queue for tickets. I also bought the rail ticket from a machine on the day of travel.

Nigel or anyone else: The most striking part of Bletchley is to see the enormous decoding machine called Bombe in action. Are demonstrations a regular event or did I just get lucky to see it all explained by an expert? Do you know of an advance schedule?

Wikipedia has a thumbnail description that gives due credit to the Polish contribution to the history, sometimes left out of other accounts. https://en.wikipedia.org/wiki/Bombe

Believe the bombe is explained on a regular schedule and it works somewhat continually.

I agree with the answer just above

Lola, When we visited it was a weekday in October during half term break. We were among the very first people in the door having taken a train timed to arrive about opening and we used the 2-for-1.

It got busier as we visited so I was happy to have arrived early. The displays in the cabinets in the Visitor Center were hard to see as the signage is small and as people gather it is almost impossible to see them. The huts and the Mansion were far better. We were saturated with info by about 1300 so made for lunch and back to town.