Let me begin by adding my voice to those who have said that Conway is the sole originator of the theory of surreal numbers, the reasons given by Andreas being critical. Moreover, based on my conversations and correspondence with Norman (Alling), with whom I did the following joint work on the surreals in the 1980s, there is no question he would concur.
(i) An Alternative Construction of Conway's Surreal Numbers, C.R. Math. Rep. Acad. Sci. Canada VIII (1986), pp. 241-46.
(ii) An Abstract Characterization of a Full Class of Surreal Numbers, C.R. Math. Rep. Acad. Sci. Canada VIII (1986), pp. 303-8.
(iii) Sections 4.02 and 4.03 of Norman Alling’s Foundations of Analysis Over Surreal Number Fields, North-Holland Publishing Co., Amsterdam, 1987.
My principal reason for writing this answer is to help clarify the relation between Conway's (1976) work, Norman’s work of 1962, and some work Norman and I did independently in the early 1980s making use of Norman’s work of 1962, and to correct the mistaken characterization of the construction of the surreals contained in Norman’s book as described by Alec in his question.
In his paper of 1962, Norman proves the existence of a real-closed field that is an $\eta_\alpha$-set of power $\aleph_\alpha$, whenever $\aleph_\alpha$ is regular and satisfies another natural set-theoretic condition, thereby providing an affirmative answer to a question of Erdös, Gillman and Henriksen that arose from their work on rings of continuous functions. To prove the result, Norman employed a construction that is a marriage of Hahn’s (1907) celebrated construction of non-Archimedean ordered fields and Hausdorff’s (1907) equally celebrated construction of $\eta_\alpha$-sets obtained from transfinite sequences of 0s and 1s. In my opinion, Norman’s result is one of the genuinely important results of the 20th-century theory of ordered algebraic systems.
On January 3, 1983 and December 20, 1982, Norman and I respectively submitted papers for publication in which we show that an isomorphic copy of Conway’s ordered field $\mathbf{No}$ could be obtained using Norman’s just-mentioned construction or simple variations thereof. Norman did this via the union of a chain $K_{\alpha + 1}$, $\alpha \in \mathbf{On}$, of real-closed fields that are $\eta_{\alpha + 1}$-sets (constructed with minor modifications as in his (1962)) and I did it more simply using Hahn’s construction in conjunction with Custa-Dutarti’s construction of successively filling in cuts (Algebra Ordinal, Rev. Acad. Ci Madrid 48 (1954), pp. 103-145), a construction Harzheim had shown leads to various $\eta_\alpha$-sets at various levels of recursion (Beiträge zur Theorie der Ordnungstypen, lnsbesondere der $\eta_\alpha$-Mengen, Math. Ann. 154 (1964), pp. 116-134). In our respective papers, we also introduced analogous constructions for families of isomorphic copies of distinguished subfields of ${\bf{No}}$ that are real-closed fields that are $\eta_\alpha$-sets. Norman’s paper appeared as Conway’s Field of Surreal Numbers, Trans. Am. Math Soc. 287, (1985), pp. 365-386, and a portion of my paper, which was initially submitted to Fund. Math., eventually appeared six years later as An Alternative Construction of Conway's Ordered Field ${\bf{No}}$, Algebra Universalis, 25 (1988), pp. 7-16. Another portion of that work appeared in my Absolutely Saturated Models, Fund. Math., 133 (1989), pp. 39-46. While I was waiting to hear back from the journal, I sent my paper to Norman, who I did not know at the time and who informed me that his paper (which I was unaware of) had been accepted for publication. Nevertheless, he believed my paper, which differed from his in various ways (including an emphasis on model theoretic connections, and the inclusion of the Custa-Dutarti cut construction which he was not familiar with), should be published, and he proposed we carry out joint work, which led to (i)-(iii) above. (i) is a treatment of the construction of the surreals based on the Custa-Dutarti cut construction, (ii) provides an axiomatiization of the surreals including its birthday structure, and (iii) is two subsections of Norman's book containing expansions of the just-said works.
As my characterization of (iii) suggests, Alec is mistaken when he asserts that in his (1987) Alling constructs the surreals as a field of formal power series (using techniques from his (1962)). What is true is that long after Norman and I introduce the surreals in that work using the Cuesta Dutari cut construction (pp. 121-127), with sums and products defined à la Conway, Norman points out (see pp. 246-247) that ${\bf{No}}$ so constructed is isomorphic to a distinguished field of formal power series. It is worth noting that while Conway was not familiar with the historical background that Norman and I drew attention to in our papers from the 1980s, Conway was aware of the relation between ${\bf{No}}$ and distinguished fields of formal power series (as is evident from Theorem 21 and the subsequent remarks from in his monograph On Numbers and Games). A simple proof of that relationship making use of ${\bf{No}}$’s simplicity hierarchy is the proof of Theorem 16, p. 1249 of the present author’s Number Systems with Simplicity Hierarchies: A Generalization of Conway’s Theory of Surreal Numbers, J. of Sym. Log. 66 (2001), pp. 1231-1258.
