This is all hypothetical but I really like this combination and could just imagine calling my daughter this sometime in the future. What do you berries think about it, and does [name]Lola[/name] sound too incomplete? What names could be used to get the nn [name]Lola[/name]?
[name]Lola[/name] is [name]WAY[/name] too insubstantial and cutesy to be a full name, but it does make a cute nickname, I guess. My favorites are starred:
[name]Eloise[/name]*
[name]Penelope[/name]
[name]Elodie[/name]*
[name]Laura[/name]
[name]Laurel[/name]*
[name]Louisa[/name]*
[name]Eulalia[/name]
[name]Lovisa[/name]
[name]Charlotte[/name]
[name]Chloe[/name]
[name]Lourdes[/name]
[name]Dolores[/name]
Elowen
[name]Ilona[/name]
[name]Liselotte[/name]
[name]Lois[/name]
[name]Lorraine[/name]
Loredana
[name]Lorelei[/name]
[name]Loretta[/name]
[name]Philomena[/name]
[name]Salome[/name]
[name]Apollonia[/name]
[name]Finola[/name]*
[name]Magnolia[/name]
[name]Iolanthe[/name]
[name]Nicola[/name]
[name]Olivia[/name]
[name]Romola[/name]
These may be too much of a stretch for you. Most go well with the middle name [name]May[/name], though.
I think it’s a cute name. I don’t think it’s incomplete but there seems to be a running theme on these boards about two syllable names ending in A being nicknames. I don’t get why if [name]Laura[/name], [name]Stella[/name] and [name]Anna[/name] are complete names [name]Lola[/name] isn’t the same…
I think you could use [name]Lola[/name] [name]May[/name] on it’s own. It’s one of my nicknames actually, and I’ve also seen people on these boards mention [name]Caroline[/name] as a way to get [name]Lola[/name].
I don’t think [name]Lola[/name] necessarily has to be a nickname, but I do think it sounds just a wee bit cutesy, especially when paired with a mn like [name]May[/name]. It’s really sweet but I’m not sure it sounds entirely sophisticated. I can picture it on a little girl in a sun dress, but maybe not so much on a grown woman in a business suit.
I think your best bet would be to use a more formal-sounding name that could be shortened to [name]Lola[/name] (IrisRose listed some great ones), with [name]May[/name] in the middle. [name]Eloise[/name] [name]May[/name] would be my favorite. She would still be [name]Lola[/name] [name]May[/name], but she would have a more ‘serious’ sounding name to put on resumes, etc.
I love [name]Charlotte[/name], [name]Louisa[/name], and [name]Laurel[/name]! 
I love it! I can picture it on a person of any age.