It sounds like you and your friend have already made considerable effort in dating the clock. I won't question the expertise of whoever suggested it "dates to about 1635" and was "almost certainly made in London," but I would keep in mind that it can be very difficult to accurately date artefacts. In this case, one concern would be that it could be a later replica of an earlier design. If you have not already done so, it would certainly be worth having the clock examined by a clock antique expert. It also can't hurt to get a second opinion if one expert has already given their advice.
Second, be cautious that the inscription may or may not have been made at the time the clock was made. It sounds like you believe that Chittel was an owner of the clock, rather than the maker. In that case, he could have owned it at any point in the seventeenth or eighteenth century, and had his name put on it at a much later date. He might not even have been born by 1635. However, my instinct says that Chittel was the clockmaker, but I don't have any evidence at this point to back that up.
Studley in Warwickshire does seem to be the most likely candidate for the place William Chittel lived. Keep in mind there are many spelling variations of this surname to look for; I wonder if Kettle is a variant of this surname. Searching the parish registers for Studley, Warwickshire indexed on FamilySearch I see:
- Thomas Kettle, William Kettle, and Martha Kettle, children of John Kettle, were baptised at Studley, Warks in 1676, 1680, and 1684 respectively.
- Thomas Chettle married Joane Phillips at Studley, Warks in 1697.
- Mary Kettle married Roger Westropp at Studley, Warks in 1688.
- Roger Kettle married Elizabeth Rawson at Studley, Warks in 1707.
As others have mentioned, wills are likely to be an important piece of the puzzle. A clock maker or owner of a clock in the seventeenth century is likely to have made a will. The primary court where Studley, Warwickshire wills were proved was Worcester Consistory Court. The higher level court would have been the Prerogative Court of Canterbury. Worcester Consistory wills are not available online, but there is an index of wills up to 1652. No obvious entry for your William Chittel in there, but he could very well have died after 1652. Further resources for tracking down wills in this court are available on the FamilySearch page Court of the Bishop of Worcester (Episcopal Consistory). I would extend your search well into the eighteenth century even though the clock is dated 1635.
There are many other records that can be checked for this period. Sadly, Protestation Returns don't seem to survive for Studley, Warwickshire, which could have been useful in verifying whether there were any Chittels living there in the early 1640s. I've checked the 1662 Hearth Tax for Studley (available on Ancestry.co.uk – subscription required) and could not see any Chittels or variants there in that year. There are other subsidy rolls and tax lists, both pre- and post-civil war, which could also be tracked down, but I'm not aware they are available online.
Lastly, I came across a clockmaker called John Hobbins of Studley, Warks in the early 18th century. He was noted in a mortgage assignment in 1718 and lease in 1736 (by that time apparently deceased). Often there were family connections with these trades, so it may be worth exploring Hobbin's will and immediate family to see if there are any links with Chittels.