I am almost 100% sure Biden wouldn't mind being bussed in....this is a man who took public transportation from his home in Delaware to the U.S. Senate regularly.
But no. The U.S. Secret Service will not allow the president to attend under those conditions. And they have the final say.
The last time an American president was permitted to overrule the Secret Service over a personal security issue was when JFK did it on November 22 1963.
Exactly. And no other US President AFAIK rode in an open car following that day.
But going back to the main topic of the guest list, I think it is perfectly reasonable that invitations will be extended directly to Heads of State, since it is a state funeral. Answering a previous question, I doubt Queen Sofia or Emperor Emeritus Akihito would even consider attending since they understand their constitutional position now and know it would not be appropriate for them to do so at a state event. Princess Beatrix seems to blur the lines sometimes; for example, she still attends state dinners in the Netherlands (which Queen Sofia has never done in Spain since her husband's abdication) and she was at Prince Philip's memorial service, but that was not a state event.
So, I would say no, I don't expect any foreign royals other than reigning monarchs and their consorts to attend. Perhaps CP Frederik might accompany Queen Margrethe II since she doesn't have a "plus one" anymore. If I am not mistaken, Prince Philippe accompanied King Albert II for example at the Queen Mother's funeral.
I am not sure how non-reigning Royal Families will be handled since their respective countries have their own Heads of State who have priority on the guest list.
At least for the Dutch royals it is quite common to take a KLM flight to their destination, for example whenever the family travels to Argentina and also when Máxima is on route for her UN work. If they have an official visit in a different part of the world the government plane is flown there in advance (with multiple stops in between) and they will fly commercial. So, they won't have any problem traveling business class on a KLM flight to London if that is the request (in other circumstances Willem-Alexander might even be a co-pilot om that flight).
I believe tommy100 and Sunnystar have clarified what the leaked memo says. No Head of State is forbidden from arriving in the UK on a non-commercial flight. If, however, they cannot fly commercially (which is the case of the POTUS or the King of Spain), they won't be able to use Heathrow to land their planes.
As Sunnystar said, Air Force One will probably use an RAF base. I still doubt the Brits would deny permission for President Biden to use Marine One, but, if they do, the President can still be transported by motorcade, which is slightly less secure, especially if Air Force One lands outside the London metro region.