I’m pretty sure it would have been Margaret Thatcher leaving power. I would have been about 7.
We didn’t have a TV, so I mostly remember some of the other kids at school (one’s Dad was ‘in the union’) singing anti-Thatcher songs in the playground, and one of the teachers making a real half-hearted token gesture effort to get them to stop, and even at the time thinking that she secretly wanted to join in.
The Apollo 11 moon landing, for certain. I was 4, and already fascinated by space and space exploration. I remember watching the TV coverage, and being just very excited about it in general.
One of the big gasoline chains (Gulf, I think) was giving away cardboard punch-out-and-assemble models of the Lunar Module. I had three copies of it – my father got one and made it for me, as did two of our neighbors. 
Other than Apollo missions, the next big national event that I remember is the 1972 presidential election. I watched the results on election night with my dad, who explained to me what a “landslide” was.
Apollo moon landing in 1969.
Reagan getting shot.
Fuzzy memories of JFK assassination and funeral. I was 6. I mostly remember wondering why the afternoon cartoons weren’t on TV. I don’t remember my parents or anyone else talking about it.
I remember seeing astronauts on TV walking on the Moon, with everyone around me telling me it was the real thing. It must have been Apollo 16 or 17, when I was 4 years old.
Beatles on Ed Sullivan
The 1956 election. All those lovely numbers ticking away on those old mechanical tally boards. Completely entrancing. Some people are just born nerds.
Not sure how much of an “event” this was, but I remember Elvis Presley singing Hound Dog on TV a few months before that. (And then the local boys starting to grow gasp sideburns!)
Apollo 13.
The first moon launch. My mother kept us up late to watch it, and fed us Coca Cola and chocolate so we wouldn’t fall asleep and miss it.
I, too, had a lot of moon-lander related paper toys.
Probably Alaska statehood. I remember coming downstairs in the morning and my mother said, “Alaska is now a state.”
I asked, “What was it before?” I was seven.
I’m not sure. I remember my father taking me outside and pointing up to the moon and telling me there were people up there. Then we went inside and looked at some of the TV coverage. That may have been the last Apollo mission or possibly the one before it.
Also in 1972 I do remember the presidential election. I remember walking home from school with three other kids and discussing who our parents were voting for. I knew Nixon was going to win because three out of four of us said Nixon.
I turned 5 in 1972.
I definitely recall Eisenhower’s second election in 1956. If you call the Davy Crockett craze a national event that’s an earlier memory.
The Beatles breaking up when I was ~5-6. They were on the cover of Life magazine and although I knew some of their songs as a preschooler, it wasn’t until then that I became aware of them as a band or as individuals with names.
If that doesn’t count, then Watergate. I remember a cartoon playing on the word “impeachment” by showing a caricature of Nixon being sent inside a giant peach… which was not far off my own understanding of the term at that time. Later on, since we were on our vacation in DC that summer, my mom and dad brought us down to the park across the street from the White House to watch Nixon’s resignation on someone else’s tiny portable TV; there was quite a crowd gathered.
Eisenhower’s first inauguration, 1953.
The election of Jimmy Carter as POTUS.
Before that I was aware that there was this thing called The Vietnam War and people went there and came back later, mostly, but I didn’t really have any idea what it was or meant, so I don’t really count that.
Iran Contra. I didn’t understand any of it but I know it was on the news and it seemed like it lasted for months and months.
The assassination of JFK. The whole school was called out to the flag pole to say the Pledge. Teachers were crying and none of us knew what was going on, even after it was explained to us. What did we know - we were third graders.
I have fuzzy memories of the Reagan-Carter election, when I was four, and pretty clear ones of Reagan’s inauguration and the assassination attempt shortly afterward. (I really liked the Reagans, especially Nancy, no doubt to the dismay of my Democratic parents. By 1984 I knew better.)
I also dimly remember hearing about the IRA hunger strikers on TV, which would have been around the same time, but I didn’t understand what it was about at all.
Not a specific event per say, but I remember at age six being pissed off when Gilligan’s Island was preempted by a speech from Nixon.