your picture didn't appear above, as I type this, but if i recall E (Vanilla) is 2nd finger, 2nd fret on the A string and then the other 3 strings pressed on the 4th fret. It's a bugger alright. You could try and barre up two frets from a D chord, of course...

E7 is a piece of piss on the other hand...

Where you can just play E7 - or just bar the 4th fret with your index finger and be careful not to hit the A string. Or add your little finger on the 7th fret on the A string.

Mind I have slender lady fingers so the 4442 formation is not a problem.

Keep on strumming.

My chord book gives the following (I've no idea if all ukes have the same string tuning, so if you're a left-hand-on-the-fretboard player, this reads from the string nearest to your nose to the one furthest away:

1st fret
4th fret
Open
2nd fret

How does that grab you?

Seemingly all the ways to play E major.

Vernier Caliper's suggestion is #1 — I'll try that when I get home, but that's a tricky chord change too.

What I have been doing is to play #3 but only fretting the top three strings and damping the bottom one with my thumb. I'll have to work out which is easiest.

#5 is superficially attractive, but you've got to whizz right down the neck of the uke and then whizz all the way back again.

After thirty years of playing guitar my fingers just go into auto-pilot when it comes to playing chords so whenever I try and play the Uke it all goes horribly wrong. However, I will get the hang of it one day.
As an aside... remember the GAS thread from a few weeks back? Well, I'm still suffering. I've been offered two choices: a 2001 Gibson Hummingbird custom shop or a 1998 Guild JF65. Both cost a packet. Both absolutely gorgeous. Which one, dear reader, should I go for?

Looking at Brookster's diagram, I play the one in the top right corner.