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From what I can find out, たっけ is just an expression that shows that you're
trying to remember something, but what does the だった part mean? I have a few
guesses, but they don't seem to match what I can understand from it.

> 今日一限から英語だったっけ
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grammar
What does the "だった" in "だったっけ" mean?
In this case, they both have almost the same meaning.

> 「今日一郎から英語だったっけ」 ~ _" **Was** English from first period today?"_
>
> 「今日一郎から英語だっけ」 ~ _" **Is** English from first period today?"_

It's also important to note that the present tense doesn't really get used
with 「っけ」 for anything other than 「だ」. It's weird to say,

> 「どう思うっけ」 ~ _" What do I think I thought I am thinking again...."_ { << This
> sounds crazy. }

But,

> 「どう思ったっけ」 ~ _" What did X think of that again...?"_

is totally normal.
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It directly translates to "it was so". Used to state a fact.

> Ex: 昨日は休みだった。(It was a holiday yesterday)

It directly translates to "was/is it?" When you say だったっけ it becomes a
question.

> Ex: 今日休みだったっけ？ (Is it a holiday today?)
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The particle **-kke** is an interrogatory particle that marks a recollection,
or the failure thereof, in the case a question word appears. The particle's
usage is often introspective: the speaker asks him/herself whether something
is recollected properly.  
Recollections are usually about the past, but it is possible to ask oneself
whether a certain event in the future, or a state that ranges from the past
into the present, is properly recalled. In that case, many languages use past
tense to mark the non-past event.  
The particle _-kke_ often appears with **-da** , or its past form **-dat-ta**
, as in the OP's example _eigo-dat-ta-kke_. In the form _-dat-ta-kke_ , past
tense **-dat-ta** emphasizes the function of introspective speech marked by
**-kke**. Translations into English can then use the past tense, and possibly
the adverb **again** :

> 何だったっけ  
>  nan-dat-ta-kke  
>  'What was (that) again?'

The OP's example must be understood in one specific way. The example is
repeated below:

> 今日（の）１限（目）から英語だったっけ？  
>  'Was it English from the first period today?'

Since the example contains 今日 **きょう** , rather than 今朝 **けさ** , we can infer
that the speaker asks **before** class. S/He is trying to recollect whether
English is the first subject of the day. In that case, the speaker tries to
recollect (remembered) data of a future event. Even though the event is about
the future, the past tense **-dat-ta** serves to emphasizes the introspective
function of **-kke**. The English translation shows that using the past tense
in these situations is not unique to Japanese.
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