Yes, hearing the tunes, or at least the rhythm of the tune will help. As for learning to read music: I think that relying on the tab to indicate pitch will work for now. You need to look at how written music notates rhythm. For the tunes you're learning, it's pretty simple, and pretty much any basic music method book (for any instrument) will tell you what you need to know, or either of these websites will cover it, likely in more detail than you need right now.

https://www.khanacademy.org/humanities/ ... ic-basics2 (video)
http://www.rhythm-in-music.com/fundamen ... hythm.html (text)

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And now there was no doubt that the trees were really moving - moving in and out through one another as if in a complicated country dance. ('And I suppose,' thought Lucy, 'when trees dance, it must be a very, very country dance indeed.')

C.S. Lewis

The tabs have been working well enough for me thus far, so I will supplement them with auditory learning, especially for purposes of determining what 3/4, 4/4, 6/8, etc. are supposed to sound like. I looked at the second site s1m0n provided in his post above mine...and have concluded that, while I am capable of learning to read music, I might well need to have things very dumbed down.

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Whistle No.1: Walton's Irish, soprano D
Whistle No. 2: green Feadóg Original, soprano D

The only way I can get the melody to come out as I hear it in my head and with the stress on the first beat is by having two voices alternating. I am not sure if I am remembering or imagining that effect. This ABC may throw some errors but works at http://www.mandolintab.net/abcconverter.php

I think the "great bell of Bow" has the root note.

X:1
T:Oranges and Lemons
N: Two voices alternating. Last bar of each line is 2/4 but not shown here to make the ABC easier to read.
M:3/4
L:1/4
K:G
P: Call and response
d/d/ B d | B G A/B/ | c A d | B G ||
w: V1.~~Or-an-ges and lem-ons Say the bells of St. Clem-ent's.
d |d B d | B G A/B/ | c A d | B G ||
w:V2.~~You owe me five far-things, Say the bells of St. Mar-tin's.
A F A | F D E/F/ | G E A | F D ||
w:V1.~~When will you pay me? Say the bells of Old Bai-ley.
A F A | D z E/F/ | G E A | D z ||
w:V2.~~When I grow rich, Say the bells of Shore-ditch.
d B d | G z A/B/ | c A d | G z ||
w:V1.~~When will that be? Say the bells of Step-ney.
A F A | D z E/F/ | G E A | D z ||
w:V2.~~I do not know, Says the great bell of Bow.

The effect on the first lines is a bit like those Breton dance tunes sung as call and response where each voice lingers on its last note giving an effect that a single melody instrument can't really replicate.

The Wikipedia page mentions it sounding like the bells in question. I don't know about those bells but to sound like church bells generally there would only be one strike for "Oran" and "say the" (the pitch of the second quaver in the melody sounds best to me). When you hear good bellringers producing a call and response effect there is a often a very subtle break in the rhythm between 'verses'. But no rests, I think.

What you have (skipped thirds and rhythmic alteration wherever there are fewer syllables) is basically what I'd have had without that Puffin book, except that I'd still have the last line in G. On which note (no pun!), I'd argue that D is only the root of the D chord and the true 'root' (not a term I'm comfortable with outside individual chords) of the song is G.

Did you mean to cut all the final bars to two beats?

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And we in dreams behold the Hebrides.

Why I teach... and where
Master of nine?

Yes, that is what I remember. In particular that all the single notes on the beat were crotchets and that there was only one beat rest at the ends of the last three lines. If only singing alternate lines a singer such as me as a kid wouldn't notice the short bar and it preserves the church bells effect. We definitely sang that melody line last. Ending on the lowest bell, the tenor, also helps the church bells effect.