Hi I use Thunderbird and have gmail accounts. Yes, they work together without issue as far as my experiences go.
Before changing emails accounts to gmail, have you tried checking your settings for your Virgin accounts in Thunderbird? To do this, Tools, Account Options and then click on your account/s in the left hand pane, working your way down from the account name do the account settings etc.
Also, you could try deleting your Virgin account and then re-set it up after checking what settings you need. You could do a search in Google for the settings you need, so as to be sure you have the right settings in the first place.
Apologies if I'm 'teaching you to suck eggs'. I appreciate you may have already tried this
Failing all that, certainly try gmail account. Good luck. Post back with any progress or otherwise. I'm sure others, probably with far more knowledge than myself, will be posting more suggestions too.