The BBC is considered fairly neutral by most independent bodies. Certainly they tend to be pro-gov't, regardless of which party is in charge, but they're not fanatical or anything about it.
They are not neutral. They are pro-UK. And as you point out, not pro-Tory or pro-Labour, it's pro-state. No they're not fanatical, they're subtle, but it's still obvious.

The real problem with the UK press is that the right-wing newspapers outnumber the left by about 3 to 1.
No, it's that the press is either left wing or right wing in the first place. It should be neither.