Short answer in 1850 you would not find a Bastardy bond. You might find a filiation order in the local petty sessions records. If these survive they would probably in theKent Archives. For a lot of areas, few survive but Kent may have more than many areas. Reay, used records of affiliation proceedings in the petty sessions for part of Kent in his research on the Blean area. He found less than 25% of unmarried mothers applied for filiation. (Microhistories 1996) Most either managed with the help of parents and other relatives or may have made informal arrangements with the father.
( longer answer, as this was a period of change )
After 1834, responsibility for bastard children placed upon mother. If she had a bastard child, she made her own arrangements or had to go into the workhouse . This was made a punitive experience. It was meant to deter women from pre marital sex, to prevent 'forced' early marriages and help lower the birth rate (it didn't and it may have led to more cases of infanticide). She could not apply to the court for a filiation order, only the responsible parish could apply. They had to apply to the Quarter sessions and needed corroboration which was hard to obtain. This was costly, more costly than 'in relief' and likely to fail for want of corroboration. There were very few cases.
1839-1844 Cases shifted back to petty sessions. Still the woman could not apply herself , still needed some corroborative evidence. The man could ask for the case to be moved to the Quarter Sessions. Still very few cases.
One of the results of the Welsh Rebecca Riots was an investigation into the working of the Bastardy clauses in the new poor law.
1844 onwards, woman could now present her own case for filliation in the petty sessions before two magistrates. The parish was not allowed to get involved at all. The woman had to go it alone which was a discouragement for many poor, illiterate women. She still had to provide some material evidence.( often others seeing the couple together after village events) The man could ask for the case to be held at the quarter sessions but had to post sureties. She had to apply within a year of the birth of the baby, unless a father who had previously been paying maintenance stopped doing so.