If I transfer my domain to Namecheap, will I lose the remaining time I have with my current registrar?
All time remaining on your domain will be transferred - you will not lose it. Expiry date of the domain will also be extended for one year.
However, if your domain expired with your old Registrar, and you renewed it with them, then we suggest you not to transfer it within 45 days of the previous expiry date since it will not add 1 more year to your domain name's expiration date, and you will lose out on the renewal fees paid to the previous Registrar.
*Note: If a .UK domain is transferred earlier than 3 months before expiration, the domain name will not be renewed and the money for the transfer will be refunded back to Namecheap account funds.