My mother wants to give me part of my inheritance now to avoid UK
inhertiance taxes. I live in the US now. She will transfer $300,000
from the UK. Will I have to pay any tax on this? What taxes would they
be? Is there a way to avoid those taxes?

Hello,
The answer may be simpler than reading the site shown below, but it is
important to understand that you must report the gift to the IRS.  I
won't venture to say if it will be taxed, it seems not, if properly
reported.  In principle, the IRS takes gift and estate tax from the
donor, if the donor is subject to US taxes.

http://www.rowbotham.com/articles/International/foreign_gifts.htm

Very near the end of the following site, the matter is also discussed.
 I have include the text from the hit since the HTML thread was so
long.

[PDF] ESTATE AND GIFT TAXATION OF NONRESIDENT ALIENS After the Economic ...
Dateiformat: PDF/Adobe Acrobat - HTML-Version
... NRAs Are Subject to US Gift Tax is on Gifts of Real Property or
Tangible ... Information regarding the identity of the foreign donor
will generally not ...
www.gtlaw.com/pub/articles/2001/chungg01a.pdf 

Obviously, this is no legal or professional advice, as the disclaimer
at the bottom of the page states.

I hope this is helpful.

If she dies within seven years of giving you the gift, it can form
part of her estate for UK inheritance tax purposes anyway:

http://www.direct.gov.uk/Topics/Money/TaxBasics/TaxBasicsArticle/fs/en?CONTENT_ID=4016736&chk=VLKdwp

Watch you don't get taxed twice.