I speeded up.

Or

I sped up.

I accelerated.

...would sound very clunky in the context of the mc's voice!

I think you’d say

The carriage sped along the road

and not

The carriage speeded along the road

But that

He sped up

Doesn’t sound right.

The OED allows ‘speeded’ as an alternative to ‘sped’ beginning in the 18th Century so I'd use it when it has 'up' following.

I think the difference is to do with whether you’re just speeding along or whether the speed is increasing or being compared with something, like what would be normal or a speed limit, for example.

To me, "speeded" or "sped" implies continuous high velocity - a truck that speeded down the road went down the road, if it sped down the next road it did that in the same manner as the first. If it speeded up or sped up, it accelerated. TBH, I see no difference between the two other than as a poor indicator of where and when you learnt or learned to speak.

That said, spell-checker is highlighting "speeded" but ignoring "sped".

G

<Added>

"a truck that speeded down the road went down the road" >> "a truck that speeded down the road went down the road very quickly"

Okay, so I conclude that 'to speed' means to go quickly, and you would say, 'he sped' , but to speed up' means to accelerate, and you would say 'he speeded up.'

you would say 'he speeded up

No, I'd usually say he accelerated. But I use both speeded and sped on a regular basis and have used both sped up and speeded up quite often also. My preference is learnt, but I often use learned... "Sped" feels more English than "speeded" (which feels American), but I think it is primarily a matter of taste.
G

I would say that 'He sped up,' is technically correct, but that you could adapt speeding to 'He started to speed up' or 'He started speeding up'.

Sped, for me.

Aha! I looked it up in Fowler, and what he says is that it depends on the basic verb:

The past tense and past participle forms are sped when the basic meaning is 'go fast' (cars sped past; he had got into the car and had sped off down the road). For other senses of the verb and for the phrasal verb speed up the past tense and past participle are normally speeded (up), e.g. in the intransitive sense 'travel at an illegal or dangerous speed': he speeded up and went through the traffic lights at 60mph; the process of reform must be speeded up.

I'm not sure quite what he means by "other sense of the verb" but clearly

"to accelerate" - "to speed up" - "he speeded up"

whereas

"to go fast" - "to speed" - "he sped".