the earliest known reference to the name Britnell is in 1366 in Aynhoe which is on the Northamptonshire/ Oxfordshire border when it is recorded as Bridenel; in 1384 in the same place it is recorded as Bretenel and in 1388/89 as Brudenell.In 1391/2 it is Bretenelle and Bretonel.
By the 15th century the name seems to have crystallised as Brudenell in Northamptonshire; In Buckinghamshire both Britnell and Brudenell variants appear in the 16th century and the earliest Britnell that we can actually trace a direct line to is Harry Bryttenell who died in Tring, Hertfordshire in 1552; he owned 50 acres or so of land in Tring Parish, plus a blacksmiths shop.Tring parish and more particularly the hamlet of Long Marston where he lived is a sort of long finger poking out in to the County of Buckinghamshire.
Bizarrely although we are Britnell when in Bucks when we move to Oxfordshire we become Brudenell in the 17th Century, reverting to Britnell when we move back to Bucks in the 18th century, after which we stayed Britnell.
As to the origin of the name, no-one really knows; the best guess is that the name is a derivative descriptive name of some one who was the son or grandson of a Breton incomer to the area,so constructed as Breton with an el diminutive ending.
As to any Celtic link, there is none obvious in the family as far back as I have gone (1552) but I suppose an immigrant Breton from Brittany might have had Celtic ancestors who moved to Brittany from England during the Anglo Saxon invasions of the fifth century, but if you mean Celtic as in Welsh or Scots or Irish the answer has to be none known and the name Britnell or Brudenell does not conform to a Celtic word forming pattern it is more Old English or Old French.
Remember that accents could vary quite a bit over small areas as the English language is only just becoming standardised in the 14th century and some letters are interchangeable for centuries to come, 't' and 'd' being prime examples, additionally vowels were often not properly sounded so that the difference between an 'e' and an 'u' would not necessarily be apparent.Try saying Britnell in a soft south Oxfordshire accent but swop the t for a d and you go from Br-tn-ll to Br-dn-ll and hence easily from Britnell to Brudnell.