The Reed in "Reed and Shaw" was Frederic/k John Reed who was admitted to the Law Society in 1840. His partner was George Ledger Shaw (d 1869). The exact address was 2 Friday Street.
Their partnership (Reed and Shaw) was dissolved on 23 Sep 1845 (London Gazette, 28 Nov 1845) at the address of 59 Friday Street, with Reed taking on all responsibility for debts due to and from the partnership.
Reed became part of Reed, Langford and Marsden at 59 Friday Street (Commercial Gazette, 16 Oct 1847; Morning Chronicle, 6 March 1855; London Gazette 14 Nov 1856). So we can assume that Reed, Langford and Marsden was the successor firm to Reed and Shaw.
The partnership of Reed, Langford and Marsden was dissolved on 11 Jan 1860 (London Gazette, 13 Jan 1860).
Reed then moved to 3 Gresham Street (London Gazette, 30 Mar 1860) in partnership with Thomas Phelps and Edward Sidgwick. The firm was called Reed, Phelps and Sidgwick. The partnership was dissolved on 8 Sep 1871 (London Gazette, 12 Sep 1871). Reed then retired to Cumberland, where he died in 1888 at the age of 79.
Phelps, Sigwick and Biddle was the successor firm. That partnership was in turn dissolved on 31 May 1905 (London Gazette, 6 Jun 1905).
Biddle, Thorne, Welsford and Sidgwick was the successor firm. That partnership was dissolved on 31 Mar 1965 (London Gazette, 23 Jul 1965).
Biddle and Co was the successor firm. In 2001, Biddle & Co merged with Pinsent Curtis to form Pinsent Curtis Biddle.
This is now known as Pinsent Masons LLP. It is a sizeable international law firm.
You may wish to ask whether Pinsent Masons has any relevant archive material from its predecessor firms or knows if/ where it could be found. Good luck!