September 1959 was an exceptionally dry and sunny month for England and Wales with a CET of 14.9 and a rainfall average of just 8.0mm (0.3 inches) (9%).

Scotland
Rainfall: 36%
Sunshine: 146%

Northern Ireland
Rainfall: 47%
Sunshine: 131%

Many places in England and Wales did not record any rain for the first three weeks and parts of East Anglia and southern Devon were completely rainless for the whole month.

From 6th to 12th, somewhere in the UK recorded a maximum of at least 80F (26.7C) with Gatwick recording 86F (30C) on the 11th.

 The period 14th August - 8th October 1959  recorded just 14.2mm for England and Wales.

September 1959 rainfall totals

Camden Square: 0.08" (4%)
Dover: 0.09" (4%)
Worthing: 0.1" (5%)
Southampton: 0.08" (3%)
Slough: 0.05" (2%)
Oxford: 0.21" (10%)
Southend: 0.13" (8%)
Ipswich: 0.02" (2%)
Lowestoft: trace (0%)
Ilfracombe: 0.40" (12%)
Scilly: 0.04" (2%)
Penzance: 0.11" (3%)
Church Stretton: 0.02" (1%)
Bath: 0.17" (7%)
Worcester: 0.18" (9%)
Birmingham: 0.17" (7%)
Mansfield: 0.07" (3%)
Buxton: 0.17" (4%)
Wakefield: 0.11" (5%)
Manchester: 0.21" (8%)
York: 0.17" (8%)
Scarborough: 0.38" (18%)
Newcastle: 0.76" (32%)
Keswick: 1.23" (21%)
Abergavenney: 0.03" (1%)
Cardiff: 0.25" (7%)
Llandudno: 0.57" (21%)
Douglas: 1.35" (32%)
Dumfries: 1.52" (41%)
Edinburgh: 0.47" (18%)
Prestwick: 1.43" (41%)
Tiree: 2.12" (51%)
Leuchars: 0.44" (18%)
Braemar: 0.64" (22%)
Fort William: 3.13" (45%)
Dyce: 0.62" (20%)
Stornoway: 2.74" (73%)
Lerwick: 1.33" (35%)
Armagh: 1.59" (54%)
Aldergrove:  1.65" (55%)
Londonderry: 1.25" (29%)

I have fond memories of this, being ten years old at the time. The 5-month "extended summer" from May to September 1959 inclusive was phenomenally good.

Legendary that is. Quite amazing - I've heard a lot about that - Still amazes me now when I look at those stats.