Finding a reliable answer is probably going to come down to finding someone at a genealogy society in Virginia who can put their hand on an article about William Ticknor/Tignor in an old journal.
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There are a few sites with information about William online. Some say he was the son of a Sir Thomas William Tignor of Sussex, England, who died in Northumberland, Virginia in 1656. Thomas is said to be the husband of Alice Gaill. But did father and son died the same year? Or are they looking at the death of William (husband of Mabel)?
Another site says that a Thomas Tigner owned land in Northumberland County in 1640. But they say that man died in 1713. If they have him confused with his possible grandfather, then maybe Wm's father
was named Thomas and
did come to the colonies.
Or do both of these people think that William's name was Thomas William?
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I found the abstracts for Lancaster County for 1652-1655 and there was no mention of a Thomas Tigner. William shows up 1 to 4 times:
Wm Tignor. Certificate for land for transportation of 6 persons into this colony. 6 April 1653. p 43
[what does this imply? who were the 6 people and where did they come from? is there any way to find out?]
Mr Tigner to pay levy on 5 tytheables to Mr Jno Cox. 24 Oct 1653. page 93
Mr Tignall, to pay levy on 7 tytheables to Mr Jno Cox. 6 Feb 1654/5. page 174
Mr Tignoll. to pay levy on 8 tytheables to Wm Leech. 7 Dec 1655. p 236
[John Cox was a/the justice in Lancaster County from at least April 1653 to December 1655.
from Virginia Colonial Abstracts, Volume 22, Beverly Fleet, 1944
image 96, page 95 in original book
https://www.familysearch.org/search/catalog/1949959?availability=El%20Dorado%20California%20Family%20History%20Center-----------
If you end up not finding some earlier authentic research and you have to do it yourself, I would ask myself these questions to work back to England:
1. When is the earliest sighting of William Tignor in Virginia, and where?
2. Who else was in that area at that time? When did they arrive, and does anyone know what part of England they came from?
3. What boat did they come on and where did it leave from?
4. Did he marry in America and was his son William born there? If not, are there matching records in England?
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Sorry not to be of any more help. I'm a little more familiar with New England records, since my wife's families were pretty early settlers there, but I know next to nothing about Virginia. So once again, I'd advise getting in touch with a genealogy society that covers Lancaster County and see what they can tell you.