i've traced his wife first. I haven't really done much for Richard except find that Christchurch is a possible dead end; at least no baptism there. (think that's probably your main problem.) Intriguingly though there's a whole line of Buddens living in the Horton/ Woodlands/ Wimborne. where Charlottes family comes from.It maybe that Richard's family moved more than once to find work. Richard himself was in Tarrant Keynston, a small village about 6 miles from Wimborne when he married )
Your couple were married in 1843. This is the entry in the marriage register for Wimborne Minster
Citation:Dorset History Centre; Dorchester, England; Reference: PE/WM, Wimborne Minister marriage register 1813-1921: 2 November 1843, Tom Budden of Keynston to Charlotte Clarke of Wimborne [https://www.ancestry.co.uk/sharing/10896399?mark=7b22746f6b656e223a2265324d4759376e326d75527443454c5950316c5a517a4252384c7669734c51735479685442474d4d4f4d383d222c22746f6b656e5f76657273696f6e223a225632227d ancestry sharing link]
(Note the abode is just where they were living at the time of the marriage)
It tells us that his father was Richard Budden, a labourer and hers was Zacheriah Clarke, a Yeoman
: The 1861 census shows us that Tom was actually born in Christchurch, Hampshire and Charlotte in Wimborne (but note, Christchurch is now in Dorset so it's parish records are actually held in the Dorset archives . It depends on website as to where you'll find them)
England and Wales Census, 1861", , FamilySearch (https://www.familysearch.org/ark:/61903/1:1:M7H6-3LH : Sun Mar 10 05:13:20 UTC 2024), Entry for Tom Budden and Charlote Budden, 1861.
So next step is to find their baptisms. To save a bit of time I looked at the Dorset OPC pages for Wimborne (these are transcripts so best also checking elsewhere, if possible images )
https://www.opcdorset.org/WimborneFiles/WimborneMinster.htm
Charlotte wasn't there and nor could I find her in the register index on Ancestry.
I then found father Zacharish with Charlotte and what was probably his wife Elizabeth on the 1841 census ( this time I used Ancestry as it has images and I find it easier. This entry should be on Family Search without images
Class: HO107; Piece: 289; Book: 5; Civil Parish: Wimborne Minster; County: Dorset; Enumeration District: 4; Folio: 5; Page: 4; Line: 1 sharing link
Zachariah and Elizabeth were in Wimborne Minster in 1851, with mother in law Jane Head . They were all born in Horton Dorset.
1851 Census, Dorset, England HO107; Piece: 1854; Folio: 528; Page 35 :Enumeration District: 1b Wimborne Minister sharing link
Next step look for a marriage between Zachariah Clarke and Elizabeth Head. And they married on 23 May 1823 (so before Charlottes birth) at Wimborne
Dorset History Centre; Dorchester, England; Reference: PE/WM Wimborne Minster Marriage Register 1813-1921 sharing link
Next trip to Horton OPC.
Zechariah Clarke baptised 19 Nov 1797 son of Zechariah and Agnes.
But there's no Elizabeth Head. She was to be found being registered in Wimborne Minster, Baptist church. She was born at Horton on 1 Oct 1803, the daughter of William and Jane Head
The National Archives (United Kingdom); Kew, Surrey, England; General Register Office: Registers of Births, Marriages and Deaths surrendered to the Non-parochial Registers Commissions of 1837 and 1857; Class Number: RG 4;; Class Number: Rg 4; Piece Number: 470 Wimborne Minster Baptist Church sharing link
So to 'conclude.
Charlotte Clarke was born in about 1824 to Zachariah (Zechariah Clarke and Elizabeth Head.
Zechariah Clarke was baptised at Horton parish church in 1797 to Zechariah and Agnes Clarke
He married Elizabeth.Head in 1823 at Wimborne.
Elizabeth Head was born in 1803 in Horton, the daughter of William and Jane.Head.