Polymyalgia Rheumatica (PMR): Given the patient's age, symptoms of progressive shoulder, hip, and wrist pain, morning stiffness, and elevated inflammatory markers (ESR 54, CRP 22), PMR is a strong consideration. The lack of response to a relatively low dose of prednisolone (15mg) may indicate the need for a higher dose, as PMR typically responds well to glucocorticoids.

Rheumatoid Arthritis (RA): Although the patient's age and symptoms could suggest RA, the lack of response to prednisolone and the absence of specific mentions of hand or foot joint involvement, symmetrical arthritis, or other classic RA features make it less likely than PMR. However, it remains a consideration.
Seronegative Spondyloarthritis: This could be a possibility, especially if there are symptoms of axial involvement or enthesitis, but the primary complaint of shoulder, hip, and wrist pain without specific mention of back pain or other spondyloarthritis features makes it less likely.
Osteoarthritis (OA): While OA could explain some of the patient's joint pain, the presence of significant morning stiffness and elevated inflammatory markers is not typical for OA, making it a less likely primary diagnosis.

Giant Cell Arteritis (GCA): Given the patient's age and elevated ESR/CRP, GCA is a critical diagnosis not to miss, as it can lead to serious complications like visual loss. Although the primary symptoms do not directly suggest GCA, the association with PMR and the potential for overlapping symptoms warrant consideration.
Malignancy-associated Arthritis: Various malignancies can present with inflammatory arthritis, and given the patient's age and lack of response to initial steroid treatment, it's essential to consider this possibility.
Infection: Septic arthritis or other infectious causes could present with joint pain and elevated inflammatory markers. Although less likely given the chronic nature of the symptoms, it's crucial not to miss such diagnoses due to their potential severity.

Relapsing Polychondritis: A rare autoimmune disorder that could present with joint pain among other symptoms, but it's less likely given the primary complaint and the absence of other characteristic features like cartilage involvement.
Amyloidosis: Could present with joint pain and elevated inflammatory markers, but it's rare and would typically be associated with other systemic symptoms or findings suggestive of amyloid deposition.
Hematologic Disorders: Certain hematologic conditions, like multiple myeloma, can present with joint pain and elevated inflammatory markers, but these would be considered rare in the differential diagnosis without other supporting evidence.