What role does big-private-money play in British & French elections?

Actually, I don’t think your analysis is correct here. In these cases, the government has passed a law granting a public right—the right to quiet at night, the right not to be harassed by an ex-husband, etc. And then you are using the government (the court) to enforce that right. That is pretty much government action. It is not a purely private matter. It is indeed government action, and the Constitution is applicable.

It’s not at all like a newspaper refusing to print your letter to the editor. It would be, if the government had passed a law saying that in certain circumstances you could bring an action to force a newspaper to print your letter. In that case, again, both parties are private, but this would definitely be a governmental restriction on free speech implicating the First Amendment.

You can campaign up until the polls close, if you want to.

I’m curious – what counts as a political ad in the U.K.?

For instance, in Canada around election time you’ll get trade unions putting out ads saying things like “teachers/nurses/public service employees provide a vital service” or what have you. That’s not political in the sense of saying “vote for party X”, but the not-so-subtle implication is “don’t vote for party Y that is campaigning on cutting services”.

I’m afraid I don’t know the precise answer to your question. The only time I can recall seeing ads about teachers/nurses/police and so on are ads directly attempting to encourage recruitment to those professions.

What I’ve never seen is ads like “Nurses are great, signed, the nursing union.”

Canada has regularly scheduled elections now too.

Generally speaking, political parties and membership organisations of any kind cannot purchase TV advertising here at any time.
During elections, and at certain other times, the parties will be allocated slots in which to make a “Party Political Broadcast”. The numbers of these which they get depends on the number of candidates they are running - I seem to recall that you had to field 50 to get one five-minute slot. The numbers of these slots is not great, and they are easily avoided if you don’t want to hear them. Usually the dread words “There now follows a party political broadcast on behalf of the XXX Party” makes most of the nation reach for the OFF switch and go to make tea. In the past they were broadcast on all channels simultaneously, but there are so many channels now that I don’t think they do that.

However, the Prime Minister is free to ignore that requirement and advise the GovGen to dissolve Parliament earlier than the four year period, even if the Gov’t still has the confidence of the House of Commons, as Prime Minister Harper did in 2008.

Relevant YouTube video.